J4 


LIBRARY   OF   THE    UNIVERSITY   OF   CALIFORNIA 


LIBRARY    OF   THE    UNIVERSITY   OF   CALIFORNIA 


LIBRARY   OF   THE    UNIVERSITY   OF   CALIFORNIA 


3 


LIBRARY   OF   THE    UNIVERSITY   OF   CALIFORNIA 


MODERN    PERSPECTIVE: 


A    TREATISE 


UPON    THE 


PRINCIPLES    AND   PRACTICE    OF    PLANE   AND 
CYLINDRICAL   PERSPECTIVE 


BY 


WILLIAM   R.   WARE, 

PROFESSOR  OF   ARCHITECTURE    IN   COLUMBIA   UNIVERSITY. 


EctifiieU  etiitiom 

PHOEBE    A.    HEARST 
ARCHiTECTU^RAL  LIBRARY 

'Ntixs  gorft 
THE   MACMILLAN   COMPANY 

LONDON :  MACMILLAN  &  CO.,  Ltd. 
1900 

All  rights  reserved 


iJ 


Copyright,  1882, 
By  JAMES  E.  OSGOOD  &  COMPANY. 

Copyright,  1900, 
By  the  MACMILLAN  COMPANY. 


TYPOGRAPHY    BY   THE    UNIVERSITY    PRESS. 
PRE88WORK     BY     THE      NORWOOD     PRESS. 


PREFACE. 


IN"  compliance  with  the  wishes  of  my  pupils,  and  in 
fulfilment  of  a  promise  made  some  time  since  to 
the  publishers  of  the  American  Architect  and  Building 
News,  I  have  collected  and  revised  a  series  of  papers 
upon  Perspective,  which  I  five  years  ago  contributed  to 
the  columns  of  that  journal,  adding  half-a-dozen  chap- 
ters and  a  dozen  pages  of  illustrations. 

A  new  treatment  of  so  old  a  theme  would  be  uncalled 
for,  did  not  even  the  more  elaborate  treatises  seem  to 
be  deficient  in  comprehensiveness  and  scientific  sim- 
plicity, while  the  practical  handbooks  fail  to  make  the 
reader  acquainted  with  some  methods  that  are  found  in 
experience  to  be  among  the  most  convenient  and  practi- 
cal of  all.  Most  of  what  I  have  to  say  is,  of  course,  in 
substance,  an  old  story;  but  it  is  a  story  which  can, 
I  think,  be  told  again  with  profit,  so  as  the  better  to  lead 
up  to  the  matter  that  is  comparatively  new.  That  I 
have  anything  to  offer  which  is  absolutely  new,  that 
I  have  in  my  explorations  found  any  field  absolutely 
untrodden  by  my  predecessors,  I  can  hardly  suppose  :  I 
am  too  used,  in  these  regions,  to  discover  the  footprints 

116258 


IV  PKEFACE. 

of  unknown  or  forgotten  pioneers  in  what  I  had  taken 
to  be  really  terra  incognita.  But  I  am  sure  that  if  the 
reader  will  accompany  me,  he  will  come  to  some  things 
that,  if  not  absolutely  novel,  are  new  to  him,  and  that 
he  will  reach  some  points  of  view  from  which  the  more 
familiar  ground  will  present  an  unaccustomed  aspect.  • 

This  discussion  of  the  subject  differs  from  that  gener- 
ally given,  in  several  particulars ;  much  greater  promi- 
nence being  assigned  to  the  phenomena  of  parallel 
planes  than  is  usual,  and  use  being  made  of  the  laws 
thus  established  to  determine  the  perspective  of  lines 
of  intersection  and  of  shadows,  —  subjects  that  seem 
hitherto  to  have  received  but  little  attention. 

The  perspective  of  divergent  lines,  also,  and  of  shad- 
ows cast  by  divergent  rays,  as  from  an  artificial  source 
of  light,  is  a  subject  that  seems  to  have  been  almost 
entirely  neglected. 

In  the  course  of  these  investigations  it  will  appear 
that  the  horizontal  plane  hardly  deserves  the  paramount  - 
importance  commonly  assigned  to  it,  and  that  the  prac- 
tice of  referring  all  constructions  to  that  plane  is  pro- 
ductive of  needless  inconvenience.  The  well-known 
method,  also,  of  points  of  distance,  and  points  of  meas- 
ures, which  is  generally  treated  as  an  auxiliary  method 
of  but  limited  serviceability,  will  be  shown  to  be  of 
universal  application,  and  to  suffice  for  the  solution  of 
almost  all  problems.     The  development  of  this  method 


PREFACE.  ■  V 

to  its  legitimate  results  leads  to  the  consistent  use  of 
the  Perspective  Plan,  rendering  unnecessary  the  con- 
struction of  the  orthographic  plan,  by  the  aid  of  which 
perspective  drawings  are  commonly  made. 

Any  treatise  on  perspective  is,  of  course,  mainly  di- 
rected to  meet  the  wants  of  the  architect;  and  the  prob- 
lems with  which  he  deals  are  free  from  most  of  the  per- 
plexities that  constantly  annoy  the  student  of  nature.- 
But  there  are  difficulties  and  apparent  anomalies  which 
confuse  the  mind  even  of  the  architectural  draughtsman, 
and  in  disposing  of  these  it  is  possible  also  to  explain 
the  discrepancies  which  are  found  to  exist  between 
sketches  made  faithfully  from  nature  and  drawings 
made  according  to  the  common  perspective  rules,  — 
discrepancies  which  have  naturally  produced  among 
artists  a  certain  disregard  and  contempt  for  the  rules 
themselves.  It  will  be  shown,  as  indeed  hardly  needs 
to  be  pointed  out,  that  in ,  drawing  from  nature  one 
works,  virtually,  not  upon  a  plane,  but  upon  a  cylinder. 
The  discussion  of  Plane  Perspective  needs  to  be  supple- 
mented, then,  by  chapters  on  Cylindrical,  or,  as  it  is 
sometimes  called,  Panoramic  Perspective;  and  an  ex- 
planation of  the  principles  and  rules  of  this  method 
show  its  results  to  be  exactly  conformable,  in  kind,  to 
those  reached  when  drawing  merely  by  the  eye.  Much 
that  I  have  to  say  is  accordingly  as  pertinent  to  the 
work  of  the  landscape  painter  or  the  historical  painter 


VI  PREFACE. 

as  to  that  of  the  architect.  Indeed,  the  questions  that 
arise  when  the  human  figure  is  to  be  drawn  in  per- 
spective mainly  concern  them. 

A  separate  chapter  discusses  certain  methods  em- 
ployed for  limiting  the  space  required  for  making  draw- 
ings in  perspective,  especially  that  of  the  late  M.  Adhe- 
mar,  —  methods  of  the  greatest  value  when,  as  in  fresco- 
painting  or  scene-painting,  the  picture  is  large  compared 
with  the  size  of  the  room  in  which  it  is  to  be  made.  I 
have  taken  the  liberty  of  considerably  modifying  the 
details  of  M.  Adhemar's  processes,  and  in  an  impor- 
tant particular  have  suggested  an  alternative  procedure, 
which,  if  not  intrinsically  preferable  to  his  own,  is  at 
least  more  in  accord  with  the  point  of  view  taken  in 
this  work. 

To  this  chapter  I  have  added  a  chapter  upon  the  in- 
terpretation of  Perspective  Drawings.  Photography  has 
given  to  the  discussion  of  this  subject  an  importance 
which  it  did  not  previously  possess,  for  it  is  often  de- 
sirable to  obtain  from  the  perspective  view  taken  by 
the  camera  the  real  proportions  or  dimensions  of  the 
object  shown.  This  is  sometimes  impossible,  sufficient 
data  not  being  furnished  by  the  picture  itself,  and  no 
other  information  being  accessible.  But  when  it  is  pos- 
sible it  is  not  difficult,  as  I  have  endeavored  to  make 
plain. 

In  all  these  chapters  I  have  avoided  a  too  formal 
method  of  demonstration,  using  a  somewhat  con  versa- 


PEEFACE.  Vll 

tional  style,  and  endeavoring  to  make  the  subject  intelli- 
gible without  employing  the  apparatus  of  theorems  and 
problems.  For  those,  however,  whose  tastes  or  habits  of 
mind  might  demand  a  more  concise  and  formal  treat- 
ment, I  have  added  a  couple  of  chapters,  in  the  first  of 
which  are  collected  the  geometrical  principles  involved 
in  the  preceding  pages,  while  tlie  second  contains,  in 
the  form  of  geometrical  problems,  solutions  not  only 
of  the  simple  questions  which  the  ordinary  practice  of 
perspective  drawing  presents,  but  of  most  of  the  more 
elementary  problems  of  Descriptive  Geometry. 

Finally,  to  meet  the  practical  needs  of  the  practical 
draughtsman,  I  have  added,  at  the  end  of  the  book,  a 
chapter  upon  The  Practical  Problem,  showing  just  how 
one  goes  to  work  to  lay  out  the  main  lines  of  a  perspec- 
tive drawing  according  to  the  system  presented  in  the 
previous  chapters.  To  these  the  student  is  referred  for 
the  further  illustration  of  matters  of  detail. 

These  last  three  chapters  are,  in  a  sense,  quite  inde- 
pendent of  what  has  preceded  them,  and  might  about  as 
well  have  come  first  as  last.  Some  readers  may  find  an 
advantage  in  first  obtaining  a  general  view  of  the  whole 
subject  by  their  aid,  before  undertaking  to  follow  the 
more  detailed  treatment  presented  in  the  body  of  the 
work. 

The  plates  that  accompany  the  text  have  taken, 
altogether,  quite  as  much  time  and  labor  in  their  prep- 


Vlll  PREFACE. 

aration  as  has  the  text  itself,  —  a  labor  which  has  been 
lightened  to  me  by  the  intelligent  co-operation  of  a 
score  of  young  men,  most  of  them  my  pupils,  who 
have,  one  after  another,  taken  their  turn  at  what  has 
seemed  a  never-ending  task.  Among  these  I  owe  my 
acknowledgments  for  something  more  than  merely  cleri- 
cal service  to  Messrs.  H.  F.  Burr,  A.  H.  Munsell,  A.  B. 
Harlow,  G.  L.  Heins,  F.  D.  Sherman,  and  G.  T.  Snelling. 
Especially  ought  I  to  mention  the  name  of  Mr.  A.  J. 
Boyden,  to  whose  intelligence  and  skill  nearly  half  the 
series  bear  witness. 

I  cannot  lay  down  my  pen  without  acknowledging 
my  indebtedness  in  this,  as  in  every  other  study  that  I 
pursued  under  his  direction,  to  my  friend  and  teacher. 
Professor  Henry  L.  Eustis,  of  the  Lawrence  Scientific 
School.  It  was  his  cordial  appreciation  and  sympathy 
that  first  encouraged  me  to  pursue  the  path  of  these 
investigations.  This  was  twenty-five  years  ago,  but  I 
have  not  forgotten  it,  and  have  borne  in  mind,  as 
every  new  point  has  presented  itself,  the  pleasure  he 
would  take  in  following  my  argument. 

WILLIAM  R.  WARE. 
Dec.  20,  1882. 

I  have  taken  advantage  of  the  opportunity  offered  hy  the  issue  of 
;i  new  edition  of  this  book  to  revise  the  text  and  to  add  in  an  Appendix 
«5ome  particulars  of  interest.  I  have  also  sliglitly  changed  the  notation, 
the  chief  alterations  being  the  use  of  the  word  liorizon  instead  of  trace 
for  the  perspective  of  the  horizon  of  a  plane,  and  the  use  of  the  word 
trace  for  its  initial  line.  W.  R.  W. 

October  31, 1900. 


MODEKN  PERSPECTIVE 


^S^ 


'^'^y^ 


CONTENTS. 


Pags 
Preface iii 


CHAPTER  I. 

THE   PHENOMEXA   OF   PERSPECTIVE   IN   NATURE. 

The  Station-Point 18 

Systems  of  Parallel  Lines 19 

Vanishing-Points 19 

Systems  of  Parallel  Planes 21 

Vanishing  Lines,  or  Horizons 21 

Solid  Objects 22 

The  Three  Principal  Rules  of  Perspective 23 

CHAPTER  IL 

THE    PHENOMENA    RELATING    TO    THE    PICTURE. 

The  Plane  of  the  Pictm-e 25 

The  Axis,  Centre,  and  Station-Point.     Perspectives  ...  26 

Fig.  1 27 

Lines  Parallel  to  the  Picture 27 

Lines  Inclined  to  the  Picture 28 

Horizontal  Lines ,28 

Vertical  Lines 29 

Plate  I.     Fig.  2 30 

Notation 30 

To  find  the  Vanishing-Point  of  a  Line,  or  System  of  Lines  32 

To  find  the  Horizon  of  a  Plane  or  System  of  Planes       .     .  33 

Lines  of  Intersection 34 


Z  CONTENTS. 

Vertical  Planes 34 

Lines  Parallel  to  the  Picture  Parallel  to  the  Horizons  of  the 

Planes  they  lie  in 35 

Sections  taken  Parallel  to  the  Picture 36 

The  Intersection  of  a  Plane  with  the  Plane  of  the  Picture  36 

Planes  Visible  only  when  below  their  Horizons      ....  37 

CHAPTER  III. 

SKETCHING   IN   PERSPECTIVE.       THE   PERSPECTIVE   PLAN.      THE 
DIVISION    OF    LINES    BY   DIAGONALS. 

Plate  II 39 

The  Vanishing-Point  of  45°,   V^ 40 

Sketching 41 

The  Perspective  Plan 41 

Bird's-eye  Views 42 

The  Division  of  Lines  .     .     .     , 43 

The  Method  of  Diagonals 43 

Halving,  Quartering,  etc 43 

Fig.  3.     Fig.  4.     Fig.  5 44 

Doubling,  Tripling,  etc.     Fig.  6 46 

Symmetrical  Division 47 

Proportional  Division 47 

Random  Lines  used  for  Proportional  Division 50 


CHAPTER  IV. 

THE   DIVISION   OF   LINES    BY   TRIANGLES. 

Plate  III 52 

The  Method  of  Diagonals 53 

The  Method  of  Triangles       54 

Point,  and  Line,  of  Proportional  Measures 55 

Fig.  7 56 

The  Division  of  Horizontal  Lines 56 

„  ,,        ,,  Vertical  Lines 57 

„  ,,        ,,  Inclined  Lines 58 

Sections  taken  Parallel  to  the  Picture 59 


CONTENTS.  3 

Random  Lines  of  Proportional  Measures 59 

Perspective  Lines  used  as  Auxiliary  Horizons 61 

Fig.  8.     Fig.  9.     Fig.  10 62 

CHAPTER  V. 

ON   THE    EXACT    DETERMINATION   OF    THE    DIRECTION   AND 
MAGNITUDE    OF    PERSPECTIVE    LINES. 

Plate  IV.     Fig.  11 65 

The  Direction  of  Lines  Parallel  to  the  Pictui'e 66 

The  Direction  of  Horizontal  Lines 66 

The  Direction  of  Inclined  Lines.     Fig.  13 67 

Points  of  Distance 68 

The  Plane  of  Measures.     The   Length  of  Lines   in  the 

Plane  of  Measures 70 

The  Scale  of  the  Drawing 71 

Fig.  12 71 

Lines  of  Vertical  Measures 72 

Lines  of  Horizontal  Measures 72 

The  Length  of  other  Lines  Parallel  to  the  Picture,  by  Scale  73 

The  Miniature  Object,  or  INIodel 73 

The  Length  of  Horizontal  Lines  Inclined  to  the  Picture,  by 

Scale 74 

Lines  of  Equal  Measures 75 

The  Points  of  Equal  Measures  the  same  as  the  Points  of 

Distance 75 

The  Method  of  the  Perspective  Plan 76 

The  Method  of  Direct  Projection 76 

Advantages  of  the  Perspective  Plan 77 

CHAPTER  VL 

THE    POSITION    OF    THE    PICTURE.       THE     OBJECT    AT    45°. 
MEASUREMENT    OF    OBLIQUELY    INCLINED    LINES. 

Position  of  the  Plane  of  the  Picture 80 

Plate  V.     Fig.  14.     Fig.  15 80 

The  Object  and  the  Picture  at  45° 81 

The  Symmetry  of  the  Vanishing-Points  and  Horizons    .     .  81 

The  Practical  Conveniences  of  this 82 


4  CONTENTS. 

Fig.  16.     Fig.  17 82 

Objects  nearer  than  the  Plane  of  Measures 83 

Obliquely  Inclined  Lines  by  Scale 84 

Manifold  Points  of  Distance 86 

The  Locus  of  Points  of  Distance 87 

The  Points  of  Distance  on  each  Horizon.     Fig.  18   .     .     .  88 

The  Locus  of  Points  of  Equal  Measures.     Fig.  19     .     .     .  89 

Kandom  Lines  of  Equal  Measures 89 

Perspective  Lines  as  Auxiliary  Horizons 90 

Equal  Measures,  Proportional  Measures,  and  Scale  Measures  91 

CHAPTER  VIL 

PARALLEL   PERSPECTIVE.       CHANGE    OF    SCALE. 

Plate  VI.     Fig.  20 94 

Parallel  Perspective 95 

Inclined  Planes  with  one  Element  Parallel  or  Perpendicular 

to  the  Picture 95 

Change  of  Scale.     Fig.  21 99 

Points  of  Half-Distance,  Quarter-Distance,  etc 101 

An  Inverse  Procedure  common 102 

Fig.  22 103 

Fig.  23 104 

Practical  Limitations  in  the  Use  of  Parallel  Perspective     .  104 

CHAPTER  VIII. 


OBLIQUE    OR    THREE-POINT    PERSPECTIVE. 


"One-Point,"  or  Parallel  Perspective     . 
"  Two-Point,"  or  Angular  Perspective  . 
"  Three- Point,"  or  Oblique  Perspective 
Plate  VII.     Fig.  24,  Fig.  25,  Fig.  26,  Fig 
The  Problem  of  the  Station-Point  and  of 

the  Picture 

Fig.  28.     Fig.  29 

To  find  the  Centre.  Fig.  30  .  .  . 
To  find  the  Station-Point.  Fig.  31  . 
Geometrical  Relations.  Fig.  32  .  . 
To  find  the  Six  Points  of  Distance.     Fig.  33 


27 
the 


Centre  of 


106 
106 
106 
107 

109 
110 
111 
112 
112 


113 


CONTENTS.  5 
CHAPTER  IX. 

THE    PERSPECTIVE    OF    SHADOWS. 

The  Phenomena  of  Shadows.     Plate  VIII 115 

Fig.  34.     Fig.  35.     Fig.  36 115 

The  Position  of  the  Sun 115 

The  Shadow  of  a  Point 116 

The  Shadow  of  a  Line 116 

Fig.  37 117 

The  Shadow  of  a  Solid  Body 117 

The  Visible  Shadow  of  a  Line;   the  Intersection  of  the 
Plane  of  its  Invisible  Shadow  with  the  Plane  on  which 

it  falls 118 

The  Horizon  of  the  Plane  of  the  Invisible  Shadow      .     .  119 

The  Sun  in  front  of  the  Spectator,  behind  the  Picture  .     .  120 

Fig  34 120 

The  Sun  behind  the  Spectator.     Fig.  35 120 

The  Vanishing-Point  of  the  Line  of  Visible  Shadow     .     .  121 

Fig.  38 121 

The  Initial  Point  of  a  Shadow 121 

The  Sun  in  the  Plane  of  the  Picture.     Fig.  36     ....  123 

The  Dividing  Line  of  Light  and  Shade 124 

Sunset 124 

Notation 126 

Shadows  on  Planes  Parallel  to  the  Lines  that  cast  them     .  127 

CHAPTER  X. 

THE  PERSPECTIVE  OF  REFLECTIONS. 

The  General  Case.     The  Reflection  of  an  Obliquely  In- 
clined Line  in  an  Obliquely  Inclined  Mirror     .     .     .     .  129 

Plate  IX.     Fig.  39.     Fig.  40.     Fig.  41 132 

The  Vanishing-Point  of  Lines  Normal  to  the  Mirror     .     .  132 

The  Horizon  of  the  Normal  Plane 133 

The  Point  where  the  Given  Line  pierces  the  Mirror       .     .  133 

The  Vanishing-Point  of  the  Reflection 134 

Special  Cases.     The  Mii-ror  Vertical,  or  at  an  Angle  w'ith 

the  Picture 135 


6  CONTENTS. 

The  Reflection  of  a  Plane  Figure 136 

The  Reflection  of  a  Point 136 

Lines  Parallel  to  the  Mirror 137 

Lines  Normal  to  the  Mirror 137 

The  Mirror  Normal  to  the  Picture 138 

The  Mirror  Parallel  to  the  Picture.     Fig.  42 139 

Reflections  in  Water.     Fig.  43 140 


CHAPTER  XL 

THE  PERSPECTIVE  OF  CIRCLES. 

The  Perspective  of  a  Circle  a  Conic  Section;  generally  an 

Ellipse 143 

Plate  X.     Fig.  44.     Fig.  45 146 

The  Centre  and  Extreme  Points.     Fig.  46 147 

To  draw  the  Perspective  Ellipse.     Fig.  47 147 

The  Pole  and  Polar  Line 148 

Ellipses  that  face  the  Centre.     Fig.  48 150 

Ellipses  that  do  not  Face  the  Centre 152 

To  obtain  the  Axis  of  an  Ellipse  by  the  Method  of  Shadows. 

Fig.  49 153 

Arcs  of  Circles,  and  Arches.     Fig.  50 155 

Concentric  Circles.  Fig.  51.  Fig.  52.  Fig.  53  .  .  .  156 
Plate  XT.     Fig.  54.     The  Circle  as  an  Ellipse.      "The 

Rotunda  of  the  Vatican  " 159 

Plate  XII.     Fig.  55.     The  Circle  as  a  Parabola.     "  The 

Hall  of  the  Biga  " 159 

Plate  XIII.    Fig.  56.     The  Circle  as  a  Hyperbola.    "  The 

Hall  of  the  Vase  " 159 


CHAPTER   XII. 

DISTORTIONS   AND   CORRECTIONS.        THE   HUMAN   FIGURE. 

Plate  XIV.     Fig.  58 160 

Horizontal  Circles 160 

Distortions  of  Circles,  Cylinders,  and  Spheres      ....  161 

Corrections.     Fig.  59 162 


CONTENTS.  7 

All  Figures  distorted 164 

The  Extent  of  the  Range  of  a  Picture 166 

Turning  the  Eyes,  or  the  Head 167 

The  Human  Figure 167 

Chinese  Shadows.     Fig.  60 167 

Historical  Pictures 167 

The  "School  of  Athens."     Fig.  63.     Fig.  64     ....  168 

The  Architectural  Background 169 

Fig.  61.     Fig.  62 170 

Guide's  "  Aurora  " 170 

Fig.  65 171 


CHAPTER  XIII. 

CYLINDRICAL,    CURVILINEAR,    OR   PANORAMIC   PERSPECTIVE. 

Plate  XV 173 

Distortions  of  Rectilinear  Objects.     Fig.  70 173 

A  Street.     Fig.  66 175 

Interiors 176 

Streets 176 

Angular  Dimensions 177 

Projection  upon  a  Cylinder.     Fig.  67 179 

Right  Lines  drawn  as  Curves 180 

Right  Lines  often  look  curved  in  Nature.     Fig.  68  .     .     .  181 

Photographs 182 

The  Extension  of  the  Range  of  the  Picture.     Fig.  69   .     .  183 

Rectifying  Sketches 184 

Geometrical  Principles 185 

Plate  XVI.     Fig.  71 186 

The  Development  of  the  Cylinder.     Fig.  72 186 

Sine-Curves 187 

Vertical  and  Inclined  Lines  and  Angles 187 

Systems  of  Sine-Curves 188 

To  construct  the  Sine-Curves.     Fig.  73 188 

Plane  Perspective  as  an  Auxiliary  Method 189 

No  Sine-Curve  necessary.     Fig.  74 189 

To  find  the  Auxiliary  Vanishing-Points.     Pig.  75    .     .     .  190 

The  use  of  a  Horizontal  Cylinder.     Fig.  76 191 


CONTENTS. 


CHAPTER  XIV. 

DIVERGENT   AND    CONVERGENT    LINES.      SHADOWS   BY 
ARTIFICIAL    LIGHT. 

Convergent  Lines 193 

The  Apex 193 

The  Apex  behind  the  Spectator 193 

The  False  Apex 194 

Plate  XVII.     Fig.  77 195 

The  Apex  in  three  positions .  195 

The  Apex  behind 195 

The  Apex  in  front 196 

The  Apex  alongside 196 

Fig.  78     ... 197 

Reflections.     Fig.  79 197 

To  find  the  False  Apex  by  Perspective.     Fig.  80      ...  197 

The  same,  by  Descriptive  Geometry 198 

Shadows  by  Artificial  Light 199 

To  find  the  Vanishing-Point  of  a  Ray  of  Artificial  Light  .  200 

Fig.  81,  «  and  & 200 

To  find  the  Horizon  of  the  Plane  of  Shadows     ....  201 

A  second  way,  the  Ray  parallel  to  the  Picture      ....  202 

Fig.  81,  c,  ^/,  and  e     ! 202 

Shadows  by  Sunlight.     Fig.  81, /and  ^r     .......  203 

A  third  way,  the  Ray  parallel  to  the  Given  Line.    Fig.  81,  h  204 

This  method  applied  to  Sunlight.     Fig.  81,  i 205 

The  Shadow  of  Vertical  Lines 206 

Plate  XVIII.     Fig.  82 206 

Shadows  of  any  Parallel  Lines.     Fig.  83 207 

Objects  visible  to  the  Personages  in  the  Picture  ....  208 

Plate  XIX.     Fig.  84.     Fig.  85 209 

Interiors  by  Candlelight 209 

The  False  Apexes 210 

The  Shadows  of  Convergent  Lines  by  Artificial  Light  .     .211 

Fig.  87.    Fig.  88 212 


■^fefi<^: 


CONTENTS.  9 
CHAPTER  XV. 

OTHER    SYSTEMS    AND    METHODS. 

Vanishing- Points  disused;  Space  economized 214 

The  Method  of  Direct  Projection, 

Plate  XX.     Fig.  89 215 

The  Mixed,  or  Common  Method. 

The  Common  Method 215 

Fig.  90.     Fig.  91 215 

Sinking  the  Perspective  Plan 217 

The  Method  of  Co-ordinates. 

Three  Rectangular  Co-ordinates 218 

Height,  Width,  and  Depth.     Fig.  92,  a  and  &      ....  219 

Vertical  Dimensions.     Fig.  92,  c 219 

The  Side  Elevation  in  Perspective.     Fig.  92,  ^   .     .     .     .  220 
The  Scales  of  Height,  Width,  and  Depth.     Fig.  93      .     .  221 
To  draw  a  Line  at  45°,  using  Fractional  Points  of  Dis- 
tance.    Fig.  94 222 

To  draw  a  Square.     Fig.  95 222 

The  Method  of  Squares. 

Squaring 222 

Fig.  96 223 

Mr.  Adhemarh  Method. 

Small-Scale  Data 224 

Reduced  Scales  of   Height  and  Width  in  an  Auxiliary- 
Plane  of  Measures 225 

Reduced  Scale  of  Depth,  and  Fractional  Points  of  Distance  225 

Plate  XXI.     Fig.  97,  a 225 

Fig.  97,  & 226 

Fig.  97,  c 228 

Vertical  Margins ...  228 


10  CONTENTS. 

Fig.  98 229 

Vertical  Margins  not  essential.     Fig.  99,  A,  B,  and  C  .     .  231 

The  Plane  of  Measures 232 

Fig.  99,  D 233 

Fig.  100 234 

Auxiliary  Directrices.     Fig.  101 235 

Circular  Mouldings.     Fig.  102 236 

Remote  Objects 237 

The  Inclined  Perspective  Plan 237 

Plate  XXII.    Fig.  103 237 

The  Sunk  Plan  preferable  to  the  Inclined  Plan    ....  239 

The  results  identical.     Fig.  104 240 

To  find  the  new  Centre  and  Ground  Line.     Fig.  105    .     .  241 

Successive  Horizontal  Planes 241 

Successive  Inclined  Planes 242 

The  results  identical 243 

Small-Scale  Data 244 

The  Horizon  need  not  be  changed 245 

Plate  XXIII.     The  Halle  aux  Bles.     Figs.  106,  107,  108, 

and  109 245 


CHAPTER  XVI. 

THE    INVERSE   PROCESS. 

Given  the  Perspective  to  find  the  real  Form  and  Dimen- 
sions       247 

Oblique  or  Three-Point  Perspective       247 

Plate  XXIV.     Fig.  110 248 

Two-Point  or  Angular  Perspective.     Fig.  Ill     ....  249 

The  Centre  given 250 

The   Vanishing- Point  of    Diagonals   given,   to  find   the 

Station-Point  and  Centre.     Fig.  112 250 

Some  real  Dimensions  being  known,  to  find  a  Point  of 

Distance 250 

The  Point  of  Distance  being  given,  to  find  the  Station- 
Point,  Centre,  etc.     Fig.  113 251 

One-Point  or  Parallel  Perspective 251 

Acute  and  Obtuse  Angles.     Fig.  114 252 


CONTENTS. 


11 


CHAPTER  XVn. 

SUMMARY.      PRINCIPLES. 

Propositions  and  Definitions. 

Plate  XXV.     Fig.  115 255 

The  Perspective  Plane 255 

The  Station-Point,  Centre,  and  Axis 256 

The  Perspective  Representation 256 

Scale  dependent  on  position  relative  to  the  Perspective 

Plane 256 

The  Plane  of  the  Picture.     Fig.  116.     Fig.  117  .     .     .     .  257 

Planes. 


Finite  and  Infinite  Planes.     Fig.  118 
The  Initial-Line,  Horizon,  and  Trace 
Parallel  Planes.     Fig.  119  .     .     .     . 
The  Angle  of  two  Planes.     Fig.  120 
Planes  parallel  to  the  Picture.     Fig.  121 
Planes  not  parallel  to  the  Picture  . 
Normal  Planes.     Fig.  122   .     .     . 

Horizontal  Planes 

Inclined  Planes.     Fig.  123  .     .     . 

Vertical  Planes 

Oblique  Planes.     Fig.  124  .     .     . 


258 
259 
259 
260 
260 
260 
261 
261 
261 
261 
262 


Lines. 

Finite  and  Infinite  Lines.     Fig.  125 262 

The  Initial-Point,  Vanishing- Point  and  its  Perspective      .  262 

Parallel  Lines.     Fig.  126 262 

The  Angle  made  by  two  Lines.     Fig.  127 263 

Lines  parallel  to  the  Picture 263 

Lines  not  parallel  to  the  Picture 264 

Normal  Lines.     Fig.  128 264 

Lines  in  the  Plane  of  the  Picture 264 

Lines  in  Planes.     Fig.  129 .  265 


12  CONTENTS. 

A  Line  in  Two  Planes.     Fig.  130 265 

Accidental  Horizons  and  Initial-Lines 266 

Lines  parallel  to  the  Horizon  of  a  Plane 266 

Points. 

Points  in  Lines.     Fig.  131 267 

Co-ordinates  of  a  Point.     Fig.  132 267 

A  Point  in  two  Planes 267 

A  Point  in  the  Plane  of  the  Picture 267 

Points  of  Distance. 

The  Point  of  Distance 267 

Its  Perspective.     Fig.  133 268 

The  Isosceles  Triangle  behind  the  Picture 269 

The  Isosceles  Triangle  in  front  of  the  Picture      ....  269 

To  find  a  Point  of  Distance.     Fig.  134.     Fig.  135  .     .     .  270 

Existing  Lines  available  as  Horizons  or  Initial  Lines         .  271 

The  Locus  of  the  Points  of  Distance.     Fig.  136  .     .     .     .  271 

Note.     Surveying 272 


CHAPTER  XYIIL 

GEOMETRICAL    PROBLEMS. 

Maxims 273 

Notation 275 

Data 276 

Plate  XXVI 278 

I.    Problems  of  Direction. 

Problems  I.  and  II.  To  find  the  Vanishing-Point  of  a 
given  Line 279 

Problem  III.  Given  a  Line  by  its  Vanishing-Point,  to 
find  the  Trace  of  Planes  normal  to  it 280 

Problem  IV.  Conversely :  Given  the  Horizon  of  a  Plane,  to 
find  the  Vanishing-Point  of  its  Axes 281 

Problem  V.  Given  a  Line  in  a  Plane,  to  find  the  Vanish- 
ing-Point of  a  second  Line  making  a  given  Angle  with  it    281 


CONTENTS.  13 

Problem  VI.     Conversely:  To  find  the  Angle  between  two 

Lines 282 

Problem  V.  a.,  Problem  VI.  a.     The  sarne,  when  one  Line 

is  parallel  to  the  picture 282 

Problem  VII.  To  find  the  Angle  between  two  Planes .  .  283 
Problem  VII.  a.  The  same,  when  the  Planes  are  normal  .  283 
Problem  VII.  b.  I'he  iame,  when  their  Horizons  are  parallel  283 
Problem  VIII.     To  find  the  Angle  between  a  Line  and  a 

Plane 283 

Problem  IX.  To  find  the  distance  of  a  Vanishing-Point 
from  the  Station-Point,  a  Point  of  Distance,  and  the 
Locus  of  the  Point  of  Distance 281 

II.    Problems  of  Dimension  and  Position. 

Problem  X.     To  cut  off  given  lengths  from  a  Perspective 

Line 285 

Problem  XL     Conversely :  To  find  the  true  length  of  a 

Perspective  Line 286 

Problem  X.  a,  and  Problem  XI.  a.     The  same,  when  the 

Line  is  parallel  to  the  Picture 287 

Problem  XI L  To  find  the  perspective  of  a  Point  .  .  .  287 
Problem  XIII.     Conversely:  To  find  the  co-ordinates  of  a 

given  Point 288 

III.   Problems  of  Planes,  Lines,  and  Points. 

Problem  XIV.     To  draw  a  Line  in  a  Plane 289 

Problem  XV.  To  pass  a  Plane  through  a  Line  ....  290 
Problem  XVI.     To  draw  in  a  Plane  a  Line  parallel  to  a 

second  Plane 291 

Problem  XVI.  a.     The  same,  when  the  second  Plane  is 

parallel  to  the  first 291 

Problem  XVI.  b.     The  same,  w^hen  the  second  Plane  is 

parallel  to  the  Picture 291 

Problem  XVI.  c.     The  same,  w  hen  the  Horizons  of  the  two 

Planes  are  parallel 292 

Problem  XVII.     To  pass  a  Line  through  a  Point  in  Space  292 

Problem  XVIII.     To  pass  a  Line  through  two  Points  .     .  293 


14  CONTENTS. 

Problem  XIX.  To  pass  a  Plane  through  three  Points .  .  293 
Problem  XX.  To  pass  a  Plane  throngh  two  Lines  .  .  .  293 
Problem  XX.  a.  The  same,  when  the  Lines  are  parallel  .  294 
Problem  XX.  b.     The  same,  when  one  Line  is  parallel  to 

the  Picture 294 

Problem  XX.  c.     The  same,  when  both  Lines  are  parallel 

to  the  Picture 294 

Problem  XXI.  To  determine  whether  two  Lines  intersect  294 
Problem  XXI.  a.     The  same,  when  one  Line  is  parallel  to 

the  Picture 295 

Problem  XXII.     To  find  the  Line  of  Intersection  of  two 

Planes 296 

Problem  XXII.  a.     T7ie  same,  when  their  Horizons  are 

parallel 296 

Problem  XXIII.     To  find  the  Point  of   Intersection  of 

three  Planes 297 

Problem  XXIV.  To  find  where  a  Line  pierces  a  Plane  .  297 
Problem  XXIV.  a.     The  same,  when  the  Line  is  parallel 

to  the  Picture 298 


IV.    Problems  of  Projection,  etc. 

Problem  XXV.     To  find  the  Projection  of  a  Point  upon  a 

Plane 298 

Problem  XXV.  a.     The  same,  when  the  Plane  is  parallel 

to  the  Picture 298 

Problem  XXVI.     To  find  the  Projection  of  a  Line  upon  a 

Plane 299 

Problem  XXVII.     To  find  the  Distance  between  a  Point 

and  a  Line 299 

Problem  XXVIII.     To   find  the   Distance  between    two 

Lines 299 

Problem  XXIX.  To  divide  a  Line  in  a  given  Ratio  .  .  300 
Problem  XXIX.  a.     The  same,  when  the  Line  is  parallel 

to  the  Picture 301 


CONTENTS.  15 


CHAPTER  XIX. 

THE   PRACTICAL    PROBLEM. 

The  Data 302 

The  Attitude  of  the  Object.     Plate  XXVII 302 

Best  at  45°,  or  parallel 303 

The  Axis  to  be  as   long,  i.  e.,  the  principal  Yanishing- 

Points  as  far  apart,  as  possible 303 

To  fix  the  Station-Point,  Centre,  Points  of  Distance,  and 

Vanishing-Point  of  Diagonals.     Fig.  137 304 

The  Object  at  45°.     Fig.  138 305 

The  Ground-Plane 305 

The  Plane  of  the  Picture 305 

Yanishing-Points  of  Oblique  Lines 306 

Traces  of  Oblique  Planes 306 

The  Yanishing-Point  of  Shadows 306 

Preliminary  Operations 307 

Parallel  Perspective 307 

The  Position  of  the  Object  horizontally 308 

The  Centre 308 

Two  Objects 309 

The  Perspective  Plan .309 

The  Ground- Line  .     .     .     c 309 

The  Sunk  Perspective  Plan 309 

The  Starting-point 310 

The  Principal  Horizontal  Lines 310 

Horizontal  Lines  inclined  to  the  Picture 310 

The  Yanishing-Point  of  Diagonals 311 

Lines  parallel  to  the  Ground-Line 311 

Several  Perspective  Plans 311 

The  Perspective  Picture 312 

The  Position  of  the  Object  vertically 312 

The  Starting-Point 312 

The  Line  of  Yertical  Measures 313 

Several  such  Lines 313 

Yertical  Dimensions 313 

Points  of  Distance  upon  Yertical  Horizons      .....  314 

Constructions  in  front  of  the  Perspective  Plane.     Fig.  133  314 


16  CONTENTS. 

Two  Lines  of  Vertical  Measures.     Fig.  140 316 

Small- Scale  Data 317 

The  Vanishing-Point  unnecessary 317 

The  Use  of  the  Diagonal 317 

Auxiliary  Horizons 317 

Oblique  Planes 318 

Choice  of  Lines  of  Measures 318 

Choice  of  Methods      . 319 

Special  Devices 319 

Mechanical  Aids.     Fig.  141 320 

APPENDIX.     NOTES. 

I.  Oblique  Perspective. 

II.  Parallel  Perspective. 

III.  45°  Perspective. 

IV.  The  Inverse  Process. 
V.  Artificial  Light. 

VI.     The  Perspective  Alphabet. 


LIST   OF   PLATES. 


I.  The  Perspective  of  Planes. 

II.  The  Perspective  Plan.     The  Use  of  Diagonals. 

III.  Division  by  Triangles. 

IV.  Directions  and  Magnitudes  by  Scale. 

V.  The  Object  at  45°.      The  Measurement  of  Inclined 

Lines. 

VI.  Parallel  Perspective.     Change  of  Scale. 

VII.  Three-Point  Perspective. 

VIII.  The  Perspective  of  Shadows. 

IX.  The  Perspective  of  Reflections. 

X.  The  Perspective  of  Circles. 

XL  The    Circle   as    an    Ellipse.     The    Rotunda  in   the 

Vatican  Museum. 

XII.  The  Circle  as  a  Parabola.     The  Hall  of  the  Biga,  in 
the  Vatican  Museum. 

XIII.  The  Circle  as  an  Hyperbola.     The  Hall  of  the  Vase, 

in  the  Vatican  Museum. 

XIV.  Distortions  and  Corrections.    The  "  School  of  Athens." 
XV.  Curvilinear  Perspective. 

XVI.  Curvilinear  Perspective. 

XVH.  The  Perspective  of  Converging  Lines. 

XVIII.  Shadows  by  Artificial  Light. 

XIX.  Shadows  by  Artificial  Light. 

XX.  The  Common  Method.     The  Method  of  Co-ordinates. 

Various  Methods. 

XXI.  Mr.  Adhemar's  Method. 

XXII.  Successive  Perspective  Plans. 

XXIII.  Successive  Perspective  Plans.    The  ^^  Halle  aux  Bles." 

XXIV.  The  Inverse  Problem. 
XXV.  General  Principles. 

XXVI.  Geometrical  Problems. 

XXVII.  The  Practical  Problem.     St.  Stephen's,  Walbrook. 

XXVIII.  Oblique,  45°,  and  Parallel  Perspective. 

XXVIII.  Illustrations  of  Notes  I.,  IL,  and  III. 
XXIX.  "  "     Note  IV. 

XXX.  «  "     XoteV. 


MODERN    PERSPECTIVE. 


CHAPTER  I. 

THE   PHENOMENA   OF   PERSPECTIVE. 

A  DRAWING  made  in  'perspective  undertakes  to  rep- 
resent objects  of  the  sliape  and  size  that  they 
actually  appear  from  a  given  point.  It  has  to  do  only 
indirectly  with  their  real  shape  and  size,  being  mainly 
concerned  with  their  apparent  outlines  and  dimensions. 
Before  trying  to  learn  how  to  draw  them,  then,  it  is  ob- 
viously desirable  to  find  out  how  they  really  look.  This 
first  chapter  will  accordingly  be  taken  up  with  consider- 
ing the  appearances  of  things,  the  phenomena  with  which 
perspective  has  to  do. 

The  things  in  question,  as  always  in  the  scientific 
study  of  form,  are  lines,  especially  straight  lines ;  plane 
figures,  especially  rectangular  figures  and  the  circle  ;  and 
solid  objects,  especially  the  sphere  and  cylinder.  The 
appearance  of  solids  bounded  by  plane  surfaces  is  deter- 
mined, of  course,  by  the  aspect  of  the  plane  figures  that 
bound  them. 

1.  Certain  phenomena  in  regard  to  the  shape  and 
size  of  these  things  are  sufficiently  obvious.     It  does 

2 


18  MODERN  PERSPECTIVE. 

not  need  to  be  pointed  out  that  everything  seems 
smaller  —  that  is  to  say,  subtends  a  smaller  visual 
angle  —  when  at  a  distance  from  the  eye  than  when 
near ;  that  consequently  the  more  distant  portions  of  a 
straight  line  seem  smaller  than  equal  divisions  near  at 
hand ;  that  in  rectangular  figures  the  farther  sides  oc- 
cupy less  space  to  the  eye  than  the  nearer  sides,  so  that 
they  present,  in  most  positions,  a  trapezoidal  rather  than 
a  rectangular  aspect,  the  sides  inclining  towards  one  an- 
other ;  that  a  circle  when  seen  in  perspective  generally 
appears  as  an  ellipse,  and  that  the  centre  of  the  circle 
does  not  occupy  the  centre  of  the  ellipse,  but  is  nearer 
to  the  farther  than  to  the  hither  edge.  These  qualitative 
determinations  are  easy  enough.  But  it  is  not  so  easy 
to  determine  the  relations  of  quantity,  to  tell  lioio  much 
smaller  a  given  distance  will  make  a  given  line  appear, 
or  just  at  what  angle  the  sides  of  the  rectangle  seem 
inclined,  and  in  what  direction  they  seem  to  run.  To 
determine  these  things  with  exactness  is  the  chief  object 
of  these  methods,  —  an  object  to  be  reached  through  the 
study  of  another  class  of  phenomena,  the  appearances 
not  of  limited  and  finite  lines  and  planes,  but  of  lines 
and  planes  supposed  to  be  indefinitely  extended.  In- 
deed, finite  lines  and  planes  are  in  perspective  consid- 
ered merely  as  portions  of  the  indefinitely  extended 
lines  and  planes  in  which  they  lie. 

2.    The  position   of   the    spectator,  that   is    to   say, 
of  the   spectator's  eye,  is  called  the  Station 

The  station  ^  ''    ' 

^''^''  Point. 


THE   PHENOMENA   OF   PERSPECTIVE.  19 

3.  All   lines  lying  in  one  and   the  same   direction, 
and  consequently  parallel  to  each  other,  are  systems  of 
said  to  belong   to  the  same   system  of  Lnes.   ^^^'^^• 
Each  line  is  an  element  of  the  system. 

Now  if  we  imagine  the  lines  of  any  system  to  be  in- 
definitely extended  both  ways,  we  shall  encounter  the 
following  phenomena. 

4.  All  the  lines  of  a  system,  that  is,  all  lines  parallel 
to  each  other  in  space,  seem  to  converge  to-  vanishing 
wards   two   infinitely   distant   points.     These  ^°^^*^' 
points  are  called  the  vanishing  points  of  that  system  of 
lines.     They  are  180°  distant  from  each  other. 

The  vanishing  points  of  a  line  are  the  utmost  possible 
limits  of  its  apparent  extension,  even  though  infinitely 
extended.  For  a  straight  line,  although  infinitely  long, 
cannot  subtend  an  arc  of  more  than  180°;  it  cannot 
seem  more  than  a  semicircle. 

The  beams  of  the  sun,  or  the  shadows  of  clouds, 
at  sunset,  which  seem  to  separate  overhead  and  con- 
verge near  the  opposite  horizon,  afford  a  capital  in- 
stance of  ^parallel  lines  with  two  vanishing  points. 
So  also  do  parallel  lines  of  cloud,  and,  in  streets,  the 
lines  of  sidewalks,  eaves,  and  house-tops.  They  ap- 
pear as  great  circles  of  the  sphere  of  which  the  eye 
is  the  centre. 

5.  Now  what  is  very  curious  is  that  whichever  ele- 
ment of  the  system  one  looks  at  seems  straight,  the 
others,  on  both  sides,  seeming  concave  towards  it.  The 
horizon  itself,  which  seems  straight  when  one  looks  at 
it,  seems  curved  if  one  looks  up  or  down.     Other  hori- 


20  MODERN  PERSPECTIVE. 

zontal  lines,  when  regarded  witli,  reference  to  the  hori- 
zon, seem  parallel  to  it,  and  farthest  removed  from  it, 
where  they  are  nearest  the  eye,  approaching  it  at  a  con- 
stantly increasing  angle  as  they  retreat  towards  their 
vanishing  points. 

These  singular  phenomena,  though  constantly  before 
our  eyes,  are  little  noticed,  and  consequently  but  little 
known  ;  but  they  sometimes  force  themselves  upon 
the  draughtsman's  attention,  causing  much  confusion  in 
his  drawing  and  in  his  mind.  The  fact  that  most 
straight  lines,  all  indeed  except  one,  always  seem 
curved  is  the  basis  of  the  method  of  curvilinear  or 
panoramic  perspective,  which  will  form  the  subject  of  a 
subsequent  cliapter. 

6.  Either  vanishing  point  of  any  system  of  lines  may 
be  found  by  looking  in  the  direction  followed  by  the 
lines  of  that  system ;  the  vanishing  point  will  then  be 
seen  full  in  front  of  the  eye. 

That  element  of  the  system  which  passes  through 
the  eye,  or  station  point,  will  be  seen  endwise,  the 
line  appearing  as  a  point,  coinciding  with  and  cover- 
ing the  vanishing  point,  which  is  at  its  farther  ex- 
tremity.    Such  a  line  we  call  an  Optical  Line. 

7.  In  like  manner,  all  planes  parallel  to  one  another, 
and  whose  axes  accordingly  belong  to  the  same  system 
of  lines,  are  said  to  belong  to  the  same  system  of  planes. 
Each  plane  is  an  element  of  the  system.  By  the  axis  of 
a  plane  is  meant  any  line  at  right  angles,  or  perpendic- 
ular, to  it. 


THE  PHENOMENA  OF  PERSPECTIVE.  21 

8.  All  the  planes  of  a  system,  that  is,  all  planes  par- 
allel to  each  other  in  space,  seem  to  converge  systems  of 

.        parallel 

towards  an  infinitely  distant  line,  which  is  pianes. 
the  limit  of  their  utmost  extension.  A  plane,  though 
seemingly  infinitely  extended,  like  the  sea,  cannot  sub- 
tend an  arc  of  more  than  180°  in  every  direction ;  it 
cannot  seem  more  than  a  hemisphere.  Its  limiting  line 
accordingly  will  be  a  great  circle  of  the  in-  vanishing 

lines,  traces, 

finite  sphere,  of  which  the  eye,  or  station  point,  or  horizons, 
is  the  centre.     This  line  is  called  the  horizon,  or  vanish- 
ing line  of  the  system  of  planes. 

9.  The  vanishing  line  or  horizon  of  any  system  of 
planes  may  be  found  by  glancing  along  that  plane  of 
the  system  which  passes  through  the  eye.  On  looking 
in  any  direction  at  right  angles  to  the  axis  of  the 
system  of  planes,  it  is  seen  full  in  front  of  the  eye. 
That  element  of  the  system  of  planes  which  passes 
through  the  eye  is  seen  edgewise,  the  plane  appear- 
inij  as  a  line,  coverinsf  and  coiucidinf{  with  the  van- 
ishing  line,  or  horizon  of  the  system,  which  is  its 
outer  extremity.  Such  a  plane  we  call  an  Optical 
Plane. 

Such  a  line  is  the  Horizon,  which  limits  at  once  the 
plane  of  the  earth,  and  the  plane,  or  hemisphere,  of  the 
sky.  We  may  call  such  a  line  the  vanishing  line  of 
a  system  of  planes,  just  as  we  speak  of  the  vanishing 
point  of  a  system  of  lines ;  but  as  it  is  common  to  call 
any  indefinitely  extended  right  line  a  vanishing  line,  it 
is  more  convenient  to  speak  of  the  horizon  of  a  system 


22  MODERN   PERSPECTIVE. 

of  planes,  distinguishing  the  real  Horizon,  or  vanishing 
line  of  horizontal  planes,  by  a  capital  H. 

10.  Any  point  or  line  lying  in  a  right  line  passing 
through  the  eye  seems  exactly  to  cover  and  coincide 
with  the  vanishing  point  of  the  system  to  which 
the  line  belongs.  So  also  any  line,  figure,  or  surface, 
lying  in  a  plaue  passing  through  the  eye,  appears  as  a 
right  line,  and  seems  to  cover  and  coincide  with  a  por- 
tion of  the  horizon  of  the  system  to  which  the  plane 
belongs. 

11.  Vanishing  points  and  horizons,  have  to  do  only 
with  the  direction  of  lines  and  planes,  not  with  their 
position.  Hence,  objects  whose  lines  and  planes  are 
parallel  have  the  same  vanishing  points  and  horizons, 
whatever  their  position  to  the  right  or  to  the  left, 
above  or  below  the  spectator. 

12.  A  plane  surface  upon  a  solid  object  cannot  be 
Solid  objects,  sceu  uulcss  it  is  ou  the  side  of  the  object 
towards  the  horizon  of  that  plane. 

13.  It  is  obvious  that  all  systems  of  horizontal  lines 
have  their  vanishing  points  in  the  Horizon,  and  con- 
versely, that  the  Horizon  passes  through  the  vanishing 
points  of  all  systems  of  horizontal  lines.  The  same  is 
true,  of  course,  of  vertical  or  inclined  planes,  and  the 
lines  that  lie  in  them  or  are  parallel  to  them.  From 
these  considerations  we  can  frame  the  following  proposi- 
tions, which  are  the  fundamental  propositions  of  our 
system  of  perspective. 


THE   PHENOMENA   OF   PERSPECTIVE.  23 

(a)  All  lines,  or  systems  of  lines,  lying  in  or  par- 
allel to  a  system  of   planes,  have  their  van-  ^^®  ^^""f^ 

•^  ^  '  principal 

ishing  points  in  the  horizon  of  that  system.   ^"Sve.'^^''' 
Hence : — 

I.  A  line  lying  iyi  a  plane  has  its  vanishing  'point 
in  the  horizon  of  that  plane. 

Conversely : 

The  horizon  of  any  system  of  planes  passes  through 
the  vanishing  points  of  all  lines  parallel  to  them. 
Hence :  — 

II.  The  horizon  of  a  plane  passes  through  the  van- 
ishing points  of  any  tiuo  lines  that  lie  in  it,  that  is,  of 
any  two  elements  of  the  plane. 

(5)  The  hoi'izons  of  all  the  systems  of  planes  which 
can  be  passed  through  a  line,  or  parallel  to  it,  in  any 
direction,  pass  through  the  vanishing  point  of  the  sys- 
tem to  wdiich  the  line  belongs,  and  intersect  each  other 
at  that  point. 

Conversely :  — 

A  line,  or  system  of  lines,  lying  in  or  parallel  to  two 
planes,  has  its  vanishing  point  at  the  intersection  of 
their  horizons.     Hence  :  — 

III.  The  line  of  intersection  of  two  planes  has  its 
vanishing  point  at  the  intersection  of  their  horizons. 

Conversely,  if  several  systems  of  planes  are  parallel  to 
the  same  line,  all  the  lines  of  intersection  of  all  the 
planes  of  all  the  systems  will  be  parallel  to  it,  and  have 
the  same  vanishing  point. 

IV.  The  Optical  lines  and  planes  will  make  the 
same  angles  with  each  other  at  the  eye,  or  station  point, 


24  MODERN   PERSPECTIVE. 

that  the  other  lines  and  planes  of  the  respective  systems 
make  with  each  other. 

14.  The  reader  is  recommended  to  take  the  pains  not 
only  to  satisfy  himself  of  the  truth  of  these  propositions, 
which  he  will  easily  do,  but  also  to  verify  them  by 
examples,  determining  for  himself,  in  his  daily  walks,  at 
what  distant  points  in  the  earth  or  the  sky  the  vanish- 
ing points  of  different  lines  are  to  be  looked  for,  lines 
horizontal,  vertical,  or  inclined  ;  and  in  like  manner  to 
trace  the  horizons  of  the  different  planes  he  encounters 
in  roofs  or  walls,  exemplifying  these  propositions  over 
and  over  again  until  they  become  perfectly  obvious  and 
familiar. 

The  vanishing  points  of  the  eaves,  for  example,  and  of 
the  raking  cornice  or  other  steepest  line  of  a  roof,  are 
easily  found  by  looking  in  the  directions  they  pursue. 
These  two  directions  determine  the  inclination  of  the 
plane  of  the  roof  in  wliich  they  lie.  Its  horizon  is  a 
great  circle,  or  straight  line,  cutting  across  tlie  sky  from 
one  of  these  vanishing  points  to  the  other.  In  the  case 
of  two  intersecting  roofs,  the  vanishing  point  of  the  hip 
or  valley  that  marks  their  intersection  is  found  at  the 
intersection  of  their  horizons. 

15.  The  discussion  of  a  problem  in  perspective  can- 
not be  considered  complete  until  the  vanishing  point  of 
every  line  and  the  horizon  of  every  plane  has  been 
determined. 


CHAPTEK  II. 

PHENOMENA   RELATING  TO   THE  PICTURE. 

IN  the  first  chapter  we  considered  the  phenomena  of 
perspective  in  nature  ;  that  is  to  say,  certain  ap- 
pearances of  the  geometrical  lines  and  surfaces  with 
which  perspective  has  to  do. 

Let  us  now — deferring  to  the  fifth  chapter  all  ques- 
tion of  exact  magnitudes,  and  of  the  precise  determina- 
tion of  forms  —  consider  in  like  manner  tlie  principal 
phenomena,  the  main  characteristics,  of  a  perspective 
drawing. 

In  so  doing  we  will  leave  all  quantitative  determina- 
tions till  by  and  by,  and  assume,  or  guess  at,  any  dimen- 
sions or  other  data  we  may  need ;  or  determine  them  by 
judgment,  or  by  the  eye.  But  as  this  is  just  the  way 
that  such  data  are  always  determined  when  sketching, 
either  from  nature  or  from  the  imagination,  it  follows 
that  what  we  have  now  to  say  is  specially  interesting  to 
the  artist  and  the  amateur,  since  it  comprises  almost 
everything  that  he  needs  in  his  own  work. 

16:  The  picture  is  supposed  to  be  drawn  upon  a  plane 
surface  called  the  plane  of  the  picture,  and  so  ^j^^  ^^^^  ^^ 
drawn   that  if  the  picture  were  transparent,  *^«p*<'*'»^- 


26  MODERN   PERSPECTIVE. 

every  point  and  line  of  the  drawing  would  cover  and 
coincide  with  the  corresponding  points  and  lines  of 
the  objects  represented,  as  seen  from  a  given  position, 
the  Station  Point;  the  plane  of  the  picture  being  at  a 
given  distance  and  in  a  given  direction. 

17.  The  distance  and  direction  of  the  picture  are 
taken  upon  a  line  passing  through  the  station  point,  at 
right  angles  with  the  plane  of  the  picture.  This  line  is 
of  course  an  axis  of  that  plane  (4).  It  is  called  the 
Axis.  If  the  Axis  if^  horizontal,  the  picture  is  vertical, 
and  this  is  the  usual  position.  But  if  the  Axis  is  in- 
clined to  the  horizontal  plane,  the  plane  of  the  picture 
is  at  an  angle  with  the  vertical  direction,  as  sometimes 
happens. 

18.  The  point  in  the  plane  of  the  picture  nearest  to 

the   eye,  or  Station   Point,  S,  is  called   the 

The  Axis,  "^     ' 

sfauonPohit  Ccutre  of  the  picture,  V^.  It  is  the  point 
Perspectives.  ^\^q^q  ^^q  ^xis  plcrccs  it.  Thc  distance  from 
the  Station  Point  to  the  Centre  is  the  length  of  the 
Axis. 

The  term  Point  of  Sight  is  used  sometimes  to  des- 
ignate the  Station  Point,  sometimes  the  Centre  of 
the  Picture.  Since  it  is  thus  ambiguous,  we  will  not 
employ  it. 

19.  The  representation  in  a  perspective  drawing 
of  a  point,  or  line,  or  of  the  vanishing  point  of  a 
line  or  system  of  lines,  or  of  the  vanishing  line,  or 
horizon,  of  a  plane  or  system  of  planes,  is  called  the 
perspective  of  the  point,  or  line^  or  vanishing  point,  or 
horizon. 


PHENOMENA   RELATING   TO   THE   PICTURE. 


27 


Figure  1  represents  the  picture-plane,  PP,  as  a  trans- 
parent plane  on  which  are  drawn  the  perspectives  of  the 
lines  behind  it.  The  perspectives  of  the  lines  drawn  on 
the  vertical  plane  behind  it,  and  which  are  consequently 
parallel  to  the  plane  of  the  picture,  are  parallel  to  the 
lines  themselves,  whether  horizontal,  vertical,  or  in- 
clined, and  though  shorter,  they  are  divided  propor- 
tionally to  them.  The  perspectives  of  the  lines  at  right 
angles  to  the  plane  of  the  picture,  however,  differ  from 
them  in  both  magnitude  and  direction.  This  illustrates 
the  following  propositions. 


Fig.  1. 

20.  Lines   parallel   to    the   picture-plane,   whatever 

their  direction,  have  their  perspectives  drawn  Lines  parallel 

to  the  pic- 
parallel  to  tliem selves ;  that  is,  in  their  real  ture. 

direction.  The  magnitude  of  the  perspective  of  any 
such  line  is  less  than  that  of  the  real  line,  according 
as  the  distance  of  the  line  itself  from  the  picture  is 
greater,  but  its  parts  are  proportional  to  the  correspond- 
ing parts  of  the  line  represented. 

21.  When   such  lines   are  parallel  they  have  their 


28  MODERN   PERSPECTIVE. 

perspectives  parallel  to  each   other,  and   to   the  lines 
themselves. 

22.  The  perspectives  of  lines  not  parallel  to  the  plane 
Lines  inclined  ^^  ^^^^  picturc  are  not  parallel  to  the  lines 
to  the  picture,  thcmselves,  nor  to  each  other,  but  are  drawn 
converging  towards  a  point  which  is  the  perspective  of 
their  vanishing  point.  In  this  case,  as  the  real  lines 
seem  to  converge  towards  their  real  vanishing  point,  so 
their  perspective  representations  do  converge  towards 
the  perspective  of  their  vanishing  point. 

23.  Hence  if  the  picture-plane  be  vertical,  as  it 
Horizontal  usuallj  is,  tlic  pcrspcctivc  of  the  liorizontal 
lines.  lines  tliat  are  parallel  to  the  plane  of  the  pic- 
ture will  be  horizontal  and  parallel,  that  of  inclined 
lines  parallel  to  the  picture  will  be  parallel  and  inclined 
at  the  same  angle  with  the  lines  themselves,  and  that  of 
all  vertical  lines  will  be  drawn  vertical  and  parallel. 
All  other  systems  of  lines,  whether  horizontal  or  in- 
clined, will  have  their  perspectives  converging  to  the 
perspective  of  their  vanishing  points. 

24.  Of  the  two  vanishing  points  belonging  to  every 
system  of  lines  (4)  one  will,  in  general,  be  behind  the 
spectator,  and  one  in  front  of  him ;  this  last  will  be 
behind  the  plane  of  the  picture,  and  its  perspective  will 
be  somewhere  in  the  plane  of  the  picture  and  at  a  finite 
distance.  But  if  the  lines  of  the  system  in  question  be 
parallel  to  the  plane  of  the  picture,  the  perspective  of  both 
vanishing  points  will  be  at  an  infinite  distance  upon  it, 
in  opposite  directions ;  and  lines  drawn  to  them  will,  of 
course,  be  parallel  (20). 


PHENOMENA   RELATING   TO    THE   PICTURE.  29 

25.  It  is  often  asked  why  the  apparent  convergence 
of  vertical  lines  is  not  represented  by  the  con-  ^r^^^.^^^^ 
vergence  of  their  perspectives,  just  as  much  ^^'^*^^- 

as  that  of  horizontal  lines.  It  is,  just  as  much.  For 
it  is  only  those  horizontal  lines  which  are  inclined  to 
the  picture  whose  perspectives  are  drawn  to  a  vanishing 
point.  The  perspectives  of  horizontal  lines  parallel  to 
the  picture  are  drawn  parallel  to  themselves,  just  as  those 
of  vertical  lines  are.  And  when  the  Axis  is  inclined  so 
that  the  plane  of  the  picture  is  no  longer  vertical,  and 
vertical  lines  are  no  longer  parallel  to  it,  they,  too,  are 
drawn  converging,  one  of  their  vanishing  points,  either 
the  zenith  or  nadir,  being  now  behind  the  picture,  and 
its  perspective  at  a  finite  distance  upon  it.  This  case 
will  be  discussed  hereafter.     See  Plate  VII. 

26.  Moreover,  although  lines  parallel  to  the  picture, 
whether  vertical,  horizontal,  or  inclined,  do  seem  to  con- 
verge towards  a  distant  vanisliing  point  just  as  other  lines 
do,  it  is  not  necessary  to  represent  this  convergence,  since 
their  perspective  representations  in  the  plane  of  the  pic- 
ture also  seem  to  converge  as  they  recede  from  the  eye, 
and  in  the  same  degree,  covering  and  coinciding  with 
them.  The  perspectives  of  the  lines  are  themselves  fore- 
shortened, and  the  space  between  them  diminished  by 
distance. 

27.  To  obtain  this  effect,  however,  in  due  degree,  as, 
indeed,  to  obtain  the  just  value  of  all  other  perspective 
effects,  the  eye  of  the  spectator  must  remain  at  the 
station  point.    From  other  points  the  picture  necessarily 


30  MODERN    PERSPECTIVE. 

looks  inexact  or  distorted.  These  distortions  increase 
from  the  centre  outward ;  and  since  it  is  so  inconvenient 
as  to  be  practically  impossible  to  keep  the  eye  always  at 
the  station  point,  it  is  best,  in  order  to  keep  this  distor- 
tion within  reasonable  limits,  not  to  extend  the  picture 
more  than  60°,  i.  e.,  not  to  make  it  much  wider  than  its 
distance  from  the  eye. 

Some  other  phenomena  relating  to  perspective  draw- 
ings are  represented  in  Plate  I.,  Figure  2. 

In  this  plate,  though  a  variety  of  objects  are  indicated, 
only  one  direction  of  each  kind  is  employed.  All  the 
Plate  I.  right-hand  liorizontal  lines  belong  to  a  single 
system,  and  all  the  left-hand  lines  to  another.  This,  of 
course,  would  not  happen  to  this  extent  in  nature ;  but 
we  have  imagined,  for  simplicity's  sake,  that  in  the  scene 
represented  all  the  buildings  are  parallel,  and  that  all 
the  roofs  are  of  the  same  pitch. 

The  Centre  of  the  picture,  V^,  the  point  nearest  the 
eye  and  opposite  the  station  point,  S,  is  here  not  exactly 
in  the  middle  of  the  picture,  but  considerably  to  the 
right,  being  just  below  the  church  on  the  hill.  The 
Station  Point  is  about  six  inches  from  the  paper ;  and 
the  eye  must  of  course  occupy  this  position  in  order  to 
make  the  things  represented  appear  of  their  proper 
shape  and   size. 

28.  In  this  plate  the  following  notation  is  adopted, 
Notation.  a  notatiou  that  will  be  adhered  to  throughout 
this  treatise,  and  with  which  the  reader  should  make  him- 
self perfectly  familiar.     Each  direction  is  indicated  by  a 


PHENOMENA   RELATING    TO   THE    PICTURE. 


31 


single  letter,  the  direction  of  each  system  of  planes  by 
two  letters,  which  give  the  direction  of  two  elements 
of  the  system  ;  each  vanishing  point  by  the  letter  V, 
with  the  letter  denoting  tlie  direction  of  the  lines  to 
which  it  belongs  written  after  it ;  and  the  horizon  of 
each  set  of  planes  by  the  letter  H,  and  the  letters  de- 
noting the  plane. 

LINES  AND  THEIR  VANISHING  POINTS. 


Lines.  Their  Direction. 

C.  Horizontal  and  normal. 

Z.  Vertical.    (To  the  zenith,  or 

nadir. ) 

R  Right-hand  horizontal 
[i.e.,  horizontal  lines  going  off  to  the  right.] 

L.  Left-hand  horizontal. 

M.  Right-hand  inclined  upwards. 

M'.  "      "            "   downwards. 

N.  Left      "            "        upwards. 

N'.  "         "            "    downwards. 

P,   Q  )  (  Inclined  lines  formed  by  the 

P',  Q' )  '  intersection  of  inclined  planes. 


Their  Vanishing  Points. 
Y°     Vanishing  point  of  normal  lines, 
"the  Centre." 


V" 

V»' 
V 

Vp,  V<i 


vertical  lines. 
Right  hor.  lines. 


Left      " 
Right  incl. 


Left  incl. 


Lines  of 
Intersection 


The  optical  lines  are  lettered  E^,  L^,  C^,  etc.  C^  is 
the  Axis. 

If  there  are  several  lines  having  the  same  general  in- 
clination they '-may  be  distinguished  by  figures,  as  R\ 
Pt^,  Pi^,  etc.  Special  vanishing  points  may  be  indicated 
as  Y\  V2,  Y3,  etc. 

PLANES  AND  THEIR  VANISHING  TRACES  (OR  HORIZONS). 


Planes.  Their  Direction. 

RZ.      [Any  plane  of  the  system  which  con-    HRZ. 
tains,  or  is  parallel  to  R  and  Z, 
i.e.,  right-hand  vertical  planes.] 

LZ.      To  L  and  Z,  i.e.,  left-hand  vertical    HLZ. 
planes. 

RL.     To  R  and  L,  i  f . ,  horizontal  planes.      HRL. 

RN.     To  R  and  N,  "inclined  up  to  the  left.  HRN. 

RN'.    ToRandN'"       "    down  "        "  HRN' 

LM.     ToLandM,"        "     up       "     right.  HLM. 

LM'   ToLandM',"       "    down"        "  HLM'. 


Their  Vanishijig  Horizon. 
Horizon  of  the  right-hand  ver- 
tical planes. 

Horizon  of  the  left-hand  vertical 

planes. 
Horizon    of    horizontal    planes 

(i.e.,  The  Horizon). 
Horizon  of  the  planes  RN. 
"         "  "      RN'. 

"         "  "      LM. 


32  MODERN  PERSPECTIVE. 

29.  The  position  of  the  various  vanishing  points,  as 
well  as  the  dimensions  of  the  various  objects,  are  sup- 
posed to  be  obtained,  in  this  picture,  as  they  would  be  ob- 
tained in  a  sketch  from  nature,  or  from  the  imagination. 

If  we  place  the  eye  at  the  station  point  and  look  in 
To  find  the  the  direction  followed  by  a  system  of  lines,  we 
Point  of  a      shall  see  their  vanishing^  point  (8) ;  and  if  the 

system  of  o  i  \    /  :> 

lines.  plane  of  the   picture  is  interposed  we   shall 

see  the  perspective  of  the  vanishing  point  in  the  same 
direction,  covering  the  real  vanishing  point.  Hence  the 
perspective  of  the  vanishing  point  of  any  system  of 
lines  is  found  by  passing  through  the  station  point  an 
element  of  that  system.  The  point  where  it  pierces  the 
picture-plane  is  the  perspective  of  the  vanishing  point 
of  the  system,  coinciding  with  and  covering  the  real 
vanishing  point. 

30.  As  M  and  M'  are  equally  inclined  to  the  horizon- 
tal plane,  one  looks  up  towards  V*^  at  exactly  the  same 
angle  that  he  looks  down  towards  V**'.  V"  is  accord- 
ingly just  as  far  above  the  Horizon  as  Y^'  is  below  it. 

The  same  is  true  of  V^  and  V^'.  But  although  M 
and  N  make  the  same  angle  with  the  ground,  the  dis- 
tance of  their  vanishing  points  above  the  Horizon  is  not 
the  same.  For  the  eye  at  S,  six  inches  in  front  of  V^,is 
further  from  V^  than  from  V^  L  being  less  inclined  to 
the  plane  of  the  picture  than  R  is.  Looking  up,  then 
at  the  same  angle,  though  tlie  eye  sees  the  real  left 
hand  upper  vanishing  point  at  the  same  height  as  the 
right-hand  one  above  the  real  Horizon,  it  sees  its  per- 
spective, V^,  higher  up  on  the  paper. 


PHENOMENA  RELATING  TO  THE  PICTURE.  33 

31.  In  like  manner  the  perspective  of  the  horizon  of 
a  plane,  or  of  a  system  of  planes,  is  found  by  ^^  ^^^  ^^^ 
passing  through  the  station  point  an  element  ^syltem^of 
of  the  system.     The  line  where  it  intersects  ^^^"^^" 
the   plane   of  the   picture   is    the   perspective   of  the 
horizon  of  the  system  of  planes. 

For  if  the  eye  is  at  the  Station  Point,  and  glances 
along  the  element  of  the  system  passing  through  it, 
it  will  see  the  horizon  upon  the  plane  of  the  picture, 
covering  and  coinciding  with  the  distant  horizon.  This 
line  is  the  line  where  the  plane  passing  through  the 
eye  cuts  the  plane  of  the  picture. 

32.  And  as  the  horizon  of  a  system  of  planes  passes 
through  the  vanishing  points  of  all  the  lines  that  lie  in 
it  or  are  parallel  to  it,  so  does  the  perspective  of  the 
horizon  pass  through  the  perspective  of  their  vanish- 
ing points ;  and  as  all  lines  lyiug  in  or  parallel  to  the 
system  have  their  vanishing  points  in  tliis  horizon, 
so  do  the  perspectives  of  all  such  lines  have  their 
point  of  convergence  or  vanishing  point  in  its  per- 
spective. 

33.  The  propositions  T.,  II.,  and  III.  (13),  are  thus  as 
true  for  the  perspectives  in  the  plane  of  the  picture  as 
for  the  real  lines  and  lines,  vanishing  points  and  hori- 
zons, as  is  exemplified  over  and  over  again  in  this  plate. 
All  the  horizons  shown  pass  through  several  vanishing 
points,  and  every  vanishing  point  lies  in  the  horizon  of 
some  system  of  planes.  Every  line  which  lies  in  two 
planes,  as  most  of  these  planes  do,  has  its  vanishing 
point  in  both  horizons,  that  is,  at  their  intersection ;  and 


34  MODEEN    PERSPECTIVE. 

the  horizons  of  all  the  planes  parallel  to  any  one  of  these 
lines  intersect  each  other  at  its  vanishing  point. 

34  It  is  specially  to  be  noted  that  the  lines  of  the 
Lines  of  In-  ^^^P'^  ^^^  vallcys  lying  at  the  intersection  of 
tersection.  ^^^^  pkncs  of  the  roofs  have  their  vanishing 
points  at  the  intersection  of  the  horizons  of  these  planes. 
Thus  the  lines  P,  P',  Q,  and  Q',  being  at  the  intersec- 
tion, respectively,  of  R  N  and  L  M,  R  N'  and  L  M',  R  N 
and  L  M',  and  R  W  and  L  M,  we  have 

yp  at  the  intersection  of  H  R  M  and  H  L  M. 

Y^'     "  "  HRN'  "    HLM'. 

VQ     "  "  HRN    "     HLM'. 

VQ'     "  "  HRN'   "     HLM. 

Y^,  the  vanishing  point  of  Q,  is  off  the  paper,  being  at 

the  intersection  of  the  horizons  HRN  and  HLM';  and 

so  in  like  manner  is  that  of  Q'  at  the  intersection  of 

H  R  N'  and  HLM. 

On  most  of  the  roofs  the  planes  L  M^  and  R  N',  being 
on  the  further  side,  are  out  of  sight.  But  the  roof  in 
the  extreme  foreground  shows  all  four  slopes.  It  has 
accordingly  been  selected  for  lettering. 

35.  This  plate  shows  also  that  if  the  picture  is  verti- 

verticai      ^^^  ^^^  horizou  of  a  vertical  plane,  such  as  L  Z 

planes.       ^^  R  Z,  is  a  vcrtical  line.     For  it  must  pass 

through  the  vanishing  point,  V^,  of  the  vertical  lines  that 

lie  in  it,  and  this  point  is  tlie  infinitely  distant  zenith. 

Besides,  it  is  the  line  in  which  tliat  plane  of  the  sys- 
tem which  passes  through  the  eye  intersects  the  plane 
of  the  picture  (31)  ;  and  as  both  these  planes  are  verti- 
cal, their  intersection  must  be  vertical. 


PHENOMENA   RELATING   TO   THE   PICTURE.  35 

36.  Hence  the  hips  and  valleys,  P  and  P',  which  lie 
in  parallel  vertical  planes,  and  accordingly  have  their  van- 
ishing points  V^  and  V^'  in  the  horizon  of  the  system  to 
which  those  planes  belong,  lie  in  a  vertical  line,  one 
exactly  above  the  other.  As  P  and  P'  are  equally  in- 
clined to  the  horizontal  plane,  V^  and  V^'  as  well  as  V^ 
and  V^^',  or  V^  and  V^',  are  equally  distant  from  the 
Horizon.  These  relations  are  indeed  sufficiently  obvi- 
ous from  the  symmetry  of  the  figure. 

The  vertical  horizon  HPP'  is  not  shown  in  the  plate. 

37.  The  proposition  that  all  lines  lying  in  a  plane 
have  their  vanishing  points  somewhere  in  the  horizon 
of  that  plane,  receives  special  illustration  in  the  case  of 
the  paths  which  cross  the  flat  open  space  beyond  the 
railroad  ;  being  level,  they  have  their  vanishing  points 
on  the  horizon,  one  at  V^,  others  at  V3  and  V.  The 
ladder  lying  on  the  roof  to  the  right  has  the  vanishing 
point  of  its  sides,  which  are  supposed  to  be  made 
parallel,  at  V^  in  the  horizon  of  the  plane  of  the  roof, 
HRN'. 

This  proposition  is  very  serviceable  in  putting  in  any 
parallel  lines  on  any  plane,  as  for  instance,  in  drawing 
the  diagonal  lines  of  slating  on  the  roof  to  the  left,  the 
vanishing  points  being  shown  at  V4  and  V5  in  the  hori- 
zon of  RN,  the  plane  of  the  roof. 

38.  But  if  a  line  lying  in  a  given  plane  is  parallel 
to   the  plane  of  the  picture,  then  its  vanish- ,. 

^  ^  Lines  parallel 

ing  point,  though  still  in  the  horizon  of  the  Jur^paSiei 
plane,  will  be  at  an  infinite  distance;  and  the  of thepSes" 
line  itself,  beino-  still  directed  to  its  vanishing    ^^  ^^^' 


36  MODERN   PERSPECTIVE. 

point,  will  be  parallel  to  the  horizon.  For  lines  which 
meet  only  at  an  infinite  distance  are  parallel.  Hence 
we  have  the  general  proposition  that,  in  any  plane,  a 
line  which  is  parallel  to  the  picture  will  be  drawn 
parallel  to  the  horizon  of  the  plane. 

30.  In  the  figure  the  broken  line  ppp  is  drawn  paral- 
sectionstak-  Icl  to  tlic  plaiic  of  the  picture.     It  is  the  line 

en  parallel  to 

the  picture,  of  iuterscction  of  a  vertical  plane  parallel  to 
the  picture,  cutting  across  the  front  corner  of  the  princi- 
pal building.  In  each  plane  it  is  parallel  to  the  hori- 
zon of  that  plane. 

40.  That  this  must  be  so  follows  not  only  from  38, 
but  also  from  the  general  proposition,  that,  if  one  sys- 
tem of  parallel  planes  intersects  another  system,  their 
lines  of  intersection  are  all  parallel. 

For  a  line  lying  in  any  plane,  and  parallel  to  the 
plane  of  the  picture,  may  be  regarded  as  the  intersection 
of  that  plane  by  a  plane  parallel  to  the  picture.  But  the 
Iiorizon  of  the  system  of  planes  in  which  the  line  lies  is 
the  line  in  which  a  plane  parallel  to  those  planes  and 
passing  through  the  eye  intersects  the  plane  of  the  pic- 
ture. We  have  thus  two  inclined  planes  parallel  to  each 
other,  intersecting  two  vertical  planes  parallel  to  each 
other.  Their  intersections  are  accordingly  parallel,  and 
the  line  in  question  is  parallel  to  the  horizon  of  the 
inclined  plane  in  which  it  lies ;  and  since  it  is  parallel 
to  the  picture,  its  perspective  is  parallel  to  itself,  and 
The  intersec-  also  is  parallel  to  tlic  horizon :  Q.  E.  D. 

tion  of  any  a  i       Ti  ir 

plane  with         41.  Morcover,  II  anv  plane  oi  that  system  of 

the  plane  of  '  J    r  J 

the  picture,    plaucs  is  extended  so  as  to  cut  the  plane  of 


PHENOMENA    RELATING   TO    THE    PICTURE.  37 

the   picture,    that   intersection   is  also    parallel  to   the 
others,  and  to  the  horizon  in  question. 

42.  This  plate  illustrates  also   the  proposition   (12) 

that  in  the  case  of  solid  objects  the  plane  surfaces  by 

which  they  are  bounded  are   visible  only  on  pianes  visi- 
ble only 

the  sides  turned  towards  their  horizons  (12);  whenbeiow 

their  hori- 

they  are  visible  only  when  they  are,  so  to  ^°"^- 
speak,  helow  their  horizons.  We  see  in  the  plate  that 
the  roof  most  nearly  below  the  eye,  being  below  all  the 
horizons,  shows  all  its  slopes,  and  so  does  the  next  one 
to  the  left.  In  all  the  others,  one,  two,  or  three  planes 
disappear,  as  they  are  above  their  horizons  ;  until,  at 
last,  in  the  case  of  the  church  on  the  top  of  tlie  hill, 
which  is  above  all  the  traces,  all  the  roofs  are  out  of 
sight.  One  of  the  houses  in  the  fort  on  the  lower  hill 
shows  the  roof  LM  just  disappearing,  the  lines  L  and  M 
both  coinciding  with  the  horizon  HLM. 

It  is  to  be  observed  that  these  horizons,  being  por- 
tions of  oreat  circles,  do  not  terminate  at  the  vanishinfjj 
points  by  which  their  position  is  determined,  but  pass 
through  and  beyond  them. 

43.  The  perspective  of  a  line  or  of  a  point  is  often 
called  a  perspective  line  or  point,  or  when  speaking  of 
the  picture,  simply  a  line  or  point.  But  in  this  last 
case,  to  avoid  confusion  of  mind,  one  must  be  careful 
to  notice  whether  the  real  line  or  point  and  the  in- 
finitely distant  vanishing  point  are  spoken  of,  or  their 
representations  in  the  plane  of  the  picture. 

So  when  one  is  speaking  of  the  horizon  of  a  system 


38  MODERN   PERSPECTIVE. 

of  planes,  he  must  be  careful  to  notice  whether 
thj  infinitely  distant  line,  or  the  line  in  the  picture 
which  covers,  and  to  the  eye  coincides  with  it,  is 
intended. 


CHAPTER  III. 

SKETCHING  IN   PERSPECTIVE.      THE   PERSPECTIVE  PLAN. 
THE   DIVISION    OF   LINES   BY   DIAGONALS. 

HAVIISTG  in  tlie  first  chapter  considered  the  nature 
of  the  plienomena  with  which  perspective  draw- 
ings have  to  do,  we  examined  in  the  last  chapter  the 
aspect  of  the  drawings  themselves,  first  observing  the 
relation  which  lines  parallel  to  the  plane  of  the  picture 
bear  to  their  perspective  representations,  and  then,  in  the 
case  of  those  not  parallel  to  the  plane  of  the  picture,  the 
relation  that  the  perspective  lines  and  planes,  by  which 
the  objects  represented  are  defined,  bear  to  the  perspec- 
tive of  their  vanishing  points  and  horizons. 

Plate  II.  illustrates  almost  all  the  points  raised  in 
explaining  Plate  I. ;  the  roofs  that  are  be-  Kate  ii. 
low  their  horizons  being  all  visible,  and  those  that  are 
above  them  being  all  out  of  sight,  while  all  the  lines  of 
intersection  of  the  planes  converge  to  the  intersection  of 
their  horizons.  This  is  specially  noticeable  of  the  valleys 
of  the  main  roof  in  the  lower  picture. 

The  horizon  H  P  P'  of  the  system  of  vertical  planes  P  P 
to  wliich  these  valleys  are  parallel,  and  which  accordingly 
passes  through  their  vanisliing  points  (13,  II.),  and  which 
was  not  drawn  in  the  previous  plate,  is  here  shown. 


40  MODERN   PERSPECTIVE. 

Since  the  lines  which  indicate  the  position  of  these  val- 
leys in  the  perspective  plan  lie  in  the  same  vertical 
planes  with  the  valleys  themselves,  they  must  have  their 
vanishing  point  in  this  same  vertical  horizon  H  P  P' ;  and 
since,  like  all  the  lines  of  this  plan,  they  lie  in  a  hori- 
zontal plane,  their  vanishing  point  must  lie  in  the 
Horizon  (13,  I.)  :  it  is  therefore  to  be  sought  at  the 
intersection  of  H  P  P'  with  the  Horizon,  at  the  point 
marked  V-^  (13,  TIL).  Since  the  vertical  planes  in 
which  these  valleys  lie  are  obviously  at  an  angle  of  45^^ 
with  the  principal  vertical  planes  R  Z  and  L  Z,  this 
line  X,  whose  vanishing  point  is  at  V''^,  is  at  45"^  with 
the  lines  R  and  L.    V^  is  off  the  paper. 

44.  K  we  put  the  eye  at  the  station  point  S,  four  or 
The  Vanish-    five  iuchcs  in  frout  of  V°,  and,  looking  first  at 

ing  Point  •it 

of  ib^.  yK  and  V^  in  directions  at  right  angles   to 

each  other,  look  then  between  them  so  as  exactly  to 
divide  the  angle,  we  shall  be  looking  in  the  direction  X, 
and  shall  see  V-^  directly  in  front  of  the  eye  (6). 

We  will  retain  this  notation,  V'^,  throughout  this 
treatise,  to  denote  the  vanishing  point  of  horizontal 
lines  making  an  angle  of  45°  with  the  principal  horizon- 
tal lines,  R  and  L  ;  and  shall  call  it,  for  brevity,  the 
vanishing  point  of  45°. 

The  little  building  on  the  left  has  steeper  roofs  than 
the  other,  their  slope  being  the  same  as  that  of  the  roof 
of  the  tower.  Their  vanishing  points  are  accordingly 
Y^\  yMi^  yNi,^  g^(3^  ^^\j[q\^  are  off  the  pa;ier. 

As  the  tower  roof  is  supposed  to  slope  alike  all  round, 
the  hips  Pi  and  Pj'  lie  also  in  parallel  planes,  at  45° ; 


THE    PERSrECTIVE    PLAN.  41 

their  projection  on  the  perspective  plan  has  V^"^  for  its 
vanishing  point,  and  Y^^  and  Y^^'  lie  in  the  horizon  of 
tlie  plane  P  P',  at  eqnal  distances  above  and  below  the 
Horizon  (36). 

45.  Tlie  position  of  tliese  vanishing  points  and  hori- 
zons is  supposed  to  be  determined  just  as  the  position  of 
the  other  leading  points  in  the  picture  is  determined; 
that  is  to  say,  their  relative  position  on  the  paper  is  made 
to  correspond  to  the  relative  position  of  the  real  points 
and  vanisliing  points,  as  nearly  as  may  be,  by  the  eye, 
by  looking  first  at  the  point,  and  then  looking  for  the 
corresponding  place  on  the  paper.  The  position  of 
the  leading  vanishing  points  being  thus  determined, 
the  horizons  can  now  be  drawn  connecting  them,  and 
new  vanishing  points,  such  as  Y^  and  Y^',  determined 
by  their  intersection. 

If  the  propositions  illustrated  in  the  last  chapter  are 
borne  in  mind,  a  consistent  and  tolerably  cor-  sketching, 
rect  perspective  sketch  can  easily  be  made,  the  eye 
being  greatly  aided  in  its  estimate  of  the  relations 
of  things,  and  their  apparent  shape  and  dimensions, 
by  the  considerations  to  which  attention  is  thus  di- 
rected. The  principal  points  being  fixed  by  the  eye, 
the  other  points  are  tlien  determined,  partly  by  the 
eye,  partly  by  means  of  lines  drawn  to  the  vanishing 
points. 

46.  A  great  advantage  may  also  be  found  in  the  use 
of  a  2^crsj)cctive  plan  of  any  object  that  is  to  be  The  Perspec- 
drawn,  especially  in  sketching,  not  from  ob-  *^^®^^^'^- 


42  MODERN   PEESPECTIVE. 

jects  but  from  the  imagination.  Thus  in  the  figure 
Figures.  (Fig.  5),  although  the  main  building  could  be 
drawn  without  much  chance  of  error,  it  is  by  no  means 
so  easy  to  determine  just  where  the  tower  behind  it 
should  make  its  appearance  over  the  roof  By  complet- 
ing the  plan,  however,  as  is  done  by  dotted  lines,  its 
position  is  at  once  determined.  The  objection  that  it  is 
undesirable  to  cover  the  drawing  with  construction  lines 
may  be  entirely  met  by  drawing  the  plan  at  a  lower 
level,  as  if  it  were  the  plan  of  the  bottom  of  the  cellar, 
ten  or  twenty  feet  underground ;  and  for  the  purpose  in 
hand,  the  cellar  may  be  supposed  to  be  of  any  conven- 
ient depth,  so  as  to  get  the  plan  entirely  out  of  the 
picture,  as  is  done  in  the  figure. 

47.  This  sinking  of  the  persi:)ective  plan  has  two  inci- 
dental advantages.  In  the  first  place,  it  makes  it  prac- 
ticable to  draw  it  on  a  separate  piece  of  paper,  which 
may  be  removed  and  kept  for  use  a  second  time,  if,  as 
often  happens,  a  perspective  drawing  needs  to  be  made 
over  again.  In  the  second  place,  it  defines  the  positions 
of  things  much  more  accurately  ;  the  lines  by  whose 
intersection  the  position  of  the  vertical  lines  is  deter- 
mined cutting  each  other  more  nearly  at  right  angles. 
It  w411  be  seen  in  the  figure  that  the  lines  in  the  real 
floor-plan  cut  each  other  so  obliquely  that  it  is  not  easy 
to  tell  exactly  where  the  corners  of  the  tower  do  come. 

48.  It  follows  from  this  that  the  level  at  which  the 
Bird's  e  e  objcct  is  to  bc  showu  in  perspective  is  quite 
Views.  independent  of  the  level  chosen  for  its  plan. 
This   also   is  illustrated   in   Plate   II.,  the   same  plan 


SKETCHING   IN    PERSPECTIVE.  43 

serving  for  three  representations  of  the  building,  at  dif- 
ferent levels,  —  one  nearly  even  with  the  eye,  with  a 
bird's-eye  view  below,  and  with  w^hat  might  be  called 
a  toad's-eye  view  of  it  above.  The  same  vanishing 
points  being  employed  in  all  three  sketches,  the  pheno- 
mena pointed  out  in  the  previous  chapter,  of  the  appear- 
ance and  disappearance  of  plane  surfaces  according  as 
they  come  below  or  above  their  horizons,  are  here  again 
illustrated. 

49.  In  thus  sketching  in  perspective,  whether  from 
nature  —  that  is,  from  a  real  object  —  or  from  the  imagi- 
nation, it  will  be  found  much  easier  to  determine  verti- 
cal magnitudes  than  horizontal  ones ;  that  is  to  say,  it 
is  easy  to  determine  the  position  of  horizontal  lines,  but 
not  their  length ;  and  the  length  of  vertical  lines,  but 
not  their  position. 

In  the  sketch,  for  example,  the  position  of  the  vanishing 
points,  and  the  position  and  height  of  the  front  corner  of 
the  building  to  be  represented,  being  once  assumed  or 
determined,  other  heights,  whether  equal  or  different, 
can  easily  be  determined  by  means  of  parallel  lines 
drawn  to  the  vanishing  points.  The  height  of  an  object 
having  been  assumed  in  one  part  of  the  picture,  an  ob- 
ject of  the  same  height  can  be  put  in  anywhere  else  by 
the  employment  of  parallel  lines. 

But  though  it  is  thus  easy  to  represent  the  three 
gable-ends  in  this  sketch  as  being  of  the  same  height,  it 
is  not  so  obvious  how  to  draw  them  so  that  they  shall 
aU  seem  equally  wide. 


44  MODERN   PERSPECTIVE. 

50.  Moreover,  the  subdivision  of  the  perspective  of 
The  Division  "^^^rtical  liucs,  whether  into  equal  parts  or 
of  Lines.  accordiug  to  some  given  proportion,  presents 
no  difficulty  ;  for  the  vertical  lines  are  parallel  to  the 
picture,  and  their  perspectives  will  accordingly  be  divided 
just  as  the  lines  themselves  are  (20). 

But  while  the  division  of  vertical  lines  and  their  ap- 
parent diminution  in  size  is  easily  managed,  the  subdi- 
vision of  horizontal  and  inclined  lines  (except  those  which 
like  the  vertical  lines  are  parallel  to  the  plane  of  the 
picture)  is  a  matter  of  difficulty.  The  more  remote  divi- 
'sions  are  smaller,  but  it  is  not  clear  how  much  smaller. 

Two  methods  are  adopted  to  determine  this,  —  the 
method  of  Diagonals,  and  the  method  of  Triangles.  Let 
us  take  the  first,  first. 

The  method  of  Diagonals  is  illustrated  in  the  various 
The  Method  fig^^i'ss  of  Plate  II.  It  applics  to  parallelo- 
of  Diagonals,  g^j^^^^g  -yvhosc  pcrspcctivcs  are  given  or  assumed 
the  following  propositions  :  — 

51.  Proiwsition  1.  A  line  drawn  through  the  intersec- 
Haiving,        tion  of  the  diagonals  of  a  parallelogram,  paral- 

quartering, 

etc.  lei  to  two  of  its  sides,  bisects  the  other  sides 

and  the  parallelogram  itself. 

This  process  may  be  repeated  with  each  half,  and  the 
Fig.  3,  a.  given  figure,  or  any  line  in  it,  divided  into  2, 
4,  8,  16,  or  32  equal  parts,  etc.     See  Fig.  3,  a. 

The  application  of  this  to  the  perspective  of  a  paral- 
5'ig-  5.  lelogram  is  shown  in  Fig.  5,  where  the  left- 

hand  side  of  the  larger  building  is  thus  divided. 


THE   DIVISION   OF   LINES   BY   DIAGONALS.  45 

52.  Less  familiar  is  the  employment  of  this  principle 
to  ascertain  the  vertical  axis  of  a  tower  two  of  whose 
sides  are  given  in  perspective,  as  in  Fig.  5.  If  diagonals 
are  drawn  across  the  tower,  from  two  points  on  the 
right-hand  vertical  corner  to  points  at  the  same  levels 
on  the  left-hand  corner,  they  will  intersect  in  the  middle 
of  the  tower,  and  a  vertical  line  through  their  intersec- 
tion may  be  used  to  determine  the  apex  of  the  roof 
which  covers  it,  as  in  the  figure.  These  diagonals  lie  in 
a  vertical  plane  that  crosses  the  tower  diagonally. 

53.  This  is  the  common  way  of  dividing  a  perspective 
line  or  surface  into  halves;  and  it  is  constantly  used,  as  on 
the  left-hand  side  of  this  building,  and  on  the  right-hand 
side  of  the  building  above  (Fig.  4),  to  determine     Fig.  4. 
the  centre  line  of  a  gable,  and  the  position  of  its  apex. 

54.  It  is  obvious  that  this  furnishes  an  alternative 
method  of  determining  the  slope  of  these  roofs.  In- 
stead, that  is,  of  fixing  the  position  of  the  vanishing 
points  of  M  and  ^I',  P  and  P',  and  thus  obtaining  the 
direction  of  these  inclined  lines,  we  may  assume  at  once 
the  direction  of  any  one  of  these  lines,  say  the  nearest 
one.  The  intersection  of  this  line  with  the  central  ver- 
tical line  fixes  the  height  of  the  roof;  the  other  slope 
and  the  other  roofs  are  then  easily  drawn. 

55.  Perspective  is  full  of  these  alternative  methods, 
different  ways  of  doing  the  same  thing.  Which  way  it  is 
best  to  adopt  in  any  given  case,  depends  upon  the  nature 
of  the  case.  In  the  present  instance,  the  vanishing 
points  Y^  and  V^'  being  outside  the  picture,  the  meth- 
od of  diao'onals  is  rather  the  most  convenient. 


46  MODERN   PERSPECTIVE. 

66.  It  is  to  be  observed,  however,  that  though  V^  and 
Y^^  are  off  the  paper,  Y^  and  Y^*  are  within  easy  reach. 
It  is  generally  worth  while,  accordingly,  to  fix  the  posi- 
tion of  the  more  remote  vanishing  points,  so  as  to  deter- 
mine the  position  of  the  traces  or  horizons  that  lie  between 
them,  and  of  the  points  where  those  horizons  intersect, 
even  if  we  make  no  direct  use  of  the  vanishing  points 
themselves.  Thus,  in  the  plate,  although  the  points  Y^\ 
yNi^  yMi/^  r^^-j^  Y^^',  which  give  the  slope  of  the  roofs  of 
the  small  house  and  of  the  tower,  are  all  at  a  distance, 
the  horizons  of  the  planes  of  the  roofs  H  E  Ni,  H  L  Mj, 
HRN/,  and  H  L  M/  all  cross  the  paper,  and  their  inter- 
sections Y^*  and  Y^^'  are  close  at  hand. 

57.  Proposition  2.  If  through  the  intersection  of  the 
diagonals  a  second  line  is  drawn  parallel  to  the  other 
two  sides  of  the  parallelogram,  a  single  diagonal  suffices 
to  effect  the  subsequent  subdivisions,  as  is  exemplified  in 
Fig.  3,  b.  Fig.  3, 1),  and  on  the  left-hand  side  of  the  larger 
Fig.  6.  building  in  Fig.  6  below. 

58.  Proposition  3.  Conversely,  if  a  line  drawn  from 
Doubling  ^^®  corner  of  a  parallelogram  to  the  middle  of 
tripling,  etc.  ^^^^  ^^  ^^iQ  opposite  sidcs  be  continued  until  it 
meets  the  other  side,  prolonged,  the  length  of  that-  side, 
or  of  the  parallelogram  itself,  may  be  doubled,  and  by 
Fig.  3,  c.  a  repetition  of  the  process,  tripled,  quadrupled, 
etc.     See  Fig.  3,  c. 

This  proposition  is  of  great  use  in  perspective  draw- 
ing, as  may  be  seen  in  Fig.  5,  where  the  gabled  end  on 
the  right  is  several  times  repeated,  each  time  smaller 
tlian  before. 


THE    DIVISION    OF   LINES   BY   DIAGONALS.  47 

It  will  be  seen  that  the  gable  ends  of  the  roofs 
grow  steeper  and  steeper,  their  lines  converging,  in  fact, 
to  the  distant  vanishing  points  M  and  M'.  By  obtain- 
ing those  points,  the  accuracy  of  these  results  can  be 
tested. 

59.  Proposition  4.  If  one  side  of  a  parallelogram  be 
divided  in  any  way  at  one  end,  equal  divi-  symmetrical 

'J  '^  '         i  division. 

sions  may  be  laid   off  at  the  other   end    by  Fig.  3,  d. 
means  of  two  diagonals.     See  Ym.  3,  d. 

This  is  very  useful  in  giving  a  symmetrical  treatment 
to  a  surface  shown  in  perspective,  as  is  seen  in  the  left- 
hand  building.  Fig.  5.  The  position  and  width  of  the 
nearer  window  on  the  side  of  the  building  being  as- 
sumed, the  vertical  lines  enclosing  the  further  window 
are  easily  found. 

At  the  end  of  the  building  the  inclined  lines  of  the 
gable,  which  may  be  regarded  as  the  semi-diagonals  of 
an  unfinished  parallelogram,  answer  the  same  purpose. 
The  base  of  any  isosceles  triangle  can  be  divided  in  this 
way. 

60.  Proposition  5.    If  one  side  of  a  parallelogram  be 
divided  in  any  way,  the  adjacent  sides  may  be  proportional 
similarly  divided  into  proportional  parts,  by  ^'"^'''"• 
means  of  one  diagonal ;  and  by  using  the  other  diago- 
nals the  order  of  the  parts  may  be  reversed.   Fig.  3,e. 
See  Fig.  3,  e. 

By  this  means  any  required  division  of  a  line  given 
in  perspective   may  be  effected,  as  is  shown   in   Fig. 


48  MODERN   PERSPECTIVE. 

6,  on  the  right-hand  sides  of  both  buildings.  The  re- 
quired division  is  made  on  the  vertical  line,  and  then 
transferred  to  the  horizontal  line  by  means  of  the  diago- 
nal, the  nearest  corner  of  the  small  house  being  divided 
according  to  the  desired  position  of  the  door  and  win- 
dows, and  that  of  the  large  building  into  three  equal 
parts. 

61.  If  the  diagonal  makes  an  angle  of  45°  with  the 
adjacent  sides,  their  segments  wiU  of  course  be  not  only 
proportional,  but  equal,  each  to  each. 

In  the  perspective  plan  of  the  small  building,  for  ex- 
ample, in  which  the  diagonals  are  directed  to  V*^,  the 
"  vanishing  point  of  45°,"  and  accordingly  make  an 
angle  of  45°  with  the  sides  of  the  building,  it  appears 
that  the  window  is  just  as  far  from  the  corner  on  one 
side  as  the  farther  edge  of  the  door  is  on  the  other.  It 
appears  also  that  the  plan  of  this  building  is  just  four 
squares,  though  it  hardly  looks  so,  the  side  being  greatly 
foreshortened,  while  the  main  part  of  the  other  building 
is  just  as  broad  as  it  is  long,  comprising  nine  squares, 
each  as  large  in  plan  as  the  tower. 

62.  In  applying  this  proposition  to  a  perspective 
drawing,  the  line  on  which  these  parts  are  first  laid  off 
must  of  course  be  a  vertical  line,  or  some  other  line 
parallel  to  the  plane  of  the  picture,  as  it  is  only  in  the 
case  of  such  lines  that  the  division  of  the  perspectives  is 
proportional  to  that  of  the  lines  themselves  (20). 

63.  Proposition  6.  It  is  not  necessary  tliat  tlie  length 
of  this  line  shall  be   previously  cletermiiied.     Indeed, 


THE   DIVISION    OF   LINES   BY   DIAGONALS.  49 

it  is  more  convenient  that  it  should  not  be,  as  it  is  easier 
to  establish  a  given  ratio  of  parts  on  an  indefinite  line. 
The  equal  or  proportionate  parts  may  be  set  off  at  any 
convenient  scale,  on  any  convenient  line  that  touches 
the  end  of  the  line  to   be  divided,   and  the 

Fig.  3,  /. 

diagonal  drawn  without  completing  the  paral- 
lelogram, as  in  Fig.  3,/. 

The  division  of  the  long  wall  in  Fig.  6,  for  instance, 
is  effected  by  setting  off  three  equal  distances  upon  the 
further  corner,  just  as  well  as  by  dividing  the  near  corner 
into  three  equal  parts. 

64.  It  is  not  necessary  in  any  of  these  cases,  of  course, 
that  the  parallelogram  shall  be  a  rectangle.  The  inclined 
line  N,  for  example,  in  the  middle  of  the  upper  figure, 
Fig.  4,  is  divided  into  four  equal  parts  by  equal 
divisions  laid  off  on  the  vertical  line  that  bisects  the 
gable. 

In  these  last  propositions,  it  will  be  observed,  use  has 
been  made  of  only  half  a  parallelogram,  that  is  to  say,  of 
a  triangle. 

65.  Proposition  5  may  then  be  restated  as  follows  :  — 
If  one  side  of  a  triangle  be  divided  in  any  way,  the 

adjacent  side  may  be  divided  into  proportional  parts  by 
means  of  lines  drawn  parallel  to  these  two 

Fig.  3,  /. 

sides  and  meeting  on  the  third  side.     See  /, 
Fig.  3. 

66.  And  from  Proposition  6  we  may  derive  this  :  — 
If  from  one  end  of  a  line  there  be  set  up  an  auxiliary, 

parallel  to  the  plane  of  the  picture,  any  parts  taken  upon 


50  MODERN   PERSPECTIVE. 

the  perspective  of  this  auxiliary  may  be  transferred  to 
the  perspective  of  the  line,  in  their  true  proportions,  by 
means  of  a  third  line  joining  the  last  point  taken  on  the 
auxiliary  with  the  other  end  of  the  first  line  (65). 

67.  But  as  any  line,  drawn  in  the  picture  at  random, 
Random  ^'^^  ^^  couccived  of  as  being  the  perspective 
pV^poSnaT  ot'  ^  liiie,  which  it  exactly  covers  and  conceals, 
drawn  parallel  to  the  picture,  it  follows  that 
any  line  whatever,  touching  one  end  of  a  perspective  line, 
may  he  used  as  an  auxiliary  hy  which  to  divide  it  in  any 
required  "proportion ;  and,  the  triangle  being  completed, 
the  first  segments  of  the  broken  lines  by  which  the 
proportions  are  transferred  will  be  parallel  to  the  line 
to  be  divided  and  will  be  directed  to  its  vanishing 
point,  and  the  second  segments  will  be  actually  paral- 
lel to  the  auxiliary  line,  since  its  vanishing  point  is 
at  an  infinite  distance. 

Thus,  if  it  is  required  to  erect  six  equidistant  spikes 
upon  the  ridge  of  the  left-hand  building,  we  may  from 
either  end  of  the  ridge  draw  a  line  in  any  convenient 
direction,  and  lay  off  on  that  line  five  equal  parts,  using 
any  convenient  scale,  as  in  the  figure.  Completing  the 
triangle  and  proceeding  as  above,  we  get  the  points  of 
division  desired.  This  triangle  does  not  lie  in  a  ver- 
tical plane  but  in  an  oblique  plane,  containing  both 
the  horizontal  line  to  be  divided  and  the  auxiliary  line  ; 
this  line  is  parallel  to  the  picture,  and  is  shown  in  its 
true  direction.     Fig.  5,  Fig.  6. 

Finally,  since  any  line  whatever  that  touches  the  end 
of  the  line  to  be  divided  will  serve  this  purpose,  it  is 


THE   DIVISION   OF   LINES,   ETC.  51 

often  convenient,  instead  of  drawing  a  new  line,  to  em- 
ploy a  line  already  existing.  We  may,  for  example,  as 
in  Fig.  6,  lay  off  our  five  equal  parts  on  the  sloping 
line  of  the  further  gable,  and  obtain' the  same  points  on 
the  ridge  as  before.  This  line,  however,  is  now  con- 
ceived to  represent,  not  the  line  of  the  gable,  but  a  line 
parallel  to  the  plane  of  the  picture,  so  taken  as  exactly 
to  cover  the  line  of  the  gable,  so  that  their  perspectives 
coincide. 


CHAPTER  IV. 

THE   DIVISION   OF  LINES   BY  THE   METHOD    OF    TRIANGLES. 

THE  third  chapter  first  set  forth  the  convenience,  in 
making  a  perspective  drawing,  of  putting  into 
Plate  III.  perspective  the  plan  of  the  object  to  be  drawn, 
*'"■  ^^'  and  of  sinking  this  plan  so  far  below  the  rep- 
resentation of  the  object  as  to  get  it  quite  free  from  the 
picture.  Plate  III.,  Fig.  10,  affords  further  illustration 
of  the  use  of  the  perspective  plan.  The  plan  of  the  gate- 
house on  the  left  is  indeed  below  the  picture,  having  in 
fact  been  drawn  on  another  piece  of  paper,  and  removed, 
as  suggested  in  a  previous  paragraph  (47).  But  the  plan  of 
the  one  in  the  distance  on  the  right  is  given,  and  it  serves 
to  determine  all  the  principal  horizontal  dimensions. 
The  plan  of  the  principal  building,  the  barn  in  the  val- 
ley, is  drawn  above  it,  instead  of  below,  as  is  sometimes 
most  convenient,  especially  in  high  buildings,  in  the 
upper  parts  of  which  it  is  of  advantage  to  have  the  per- 
spective plan  near  at  hand.  It  is  often  a  convenience, 
also,  to  make  several  plans,  set  one  above  another,  taken 
at  different  levels.  In  the  plate,  for  example,  w^e  have 
first  the  plan  of  the  outline  of  the  walls,  to  determine 
the  position  of  the  doors  and  windows,  and  of  the  posts 
of  the  shed,   and   then  just  above  it  the  plan  of  the 


THE  DIVISION  OF  LINES,  ETC.  53 

eaves,  showing  their  projection,  and  the  position  of 
the  brackets  beneath  them.  As  only  the  front  part  of 
the  building  is  seen,  only  the  front  part  of  the  plan 
needs  to  be  drawn.  In  putting  in  the  eaves,  advantage 
is  taken  of  the  "  vanishing  point  of  45°,"  Y-^,  to  make 
them  equally  wide  on  each  side. 

In  this  plate  the  slope  of  the  roofs  and  gables  of  this 
building,  as  well  as  of  the  smaller  one  with  a  hipped  roof 
beyond  it,  is  indicated  by  the  same  letters  as  in  the  pre- 
vious plates,  and  their  vanishing  points  accordingly  by 
V^,  V^^',  etc.,  as  before.  The  gables  of  the  little  gate- 
houses are  so  steep  that  their  vanishing  points  are  quite 
out  of  reach ;  and  these  gables  are,  in  fact,  drawn  by  the 
method  of  diagonals,  as  described  in  the  previous  chap- 
ter (54).  The  slope  of  the  steps  is  given  by  V^^  and 
V^^',  and  tlie  trace  of  the  inclined  planes  of  the  bank  by 
H  L  Ml  and  H  L  M/.  Their  position  shows  that  the 
banks  are  a  little  steeper  than  the  roof  of  the  barn.  The 
diagonal  braces  of  the  fence  have  nearly  the  same  slope 
as  the  barn  shed,  converging  to  points  just  below  V^  and 
just  above  Y^'. 

In  this  plate  the  centre,  V^^  is  again  quite  out  of  the 
middle  of  the  j^icture.  The  station -point,  S,  the  proper 
position  of  the  eye,  is  about  six  inches  in  front  of  V*^. 

68.    The  first  of  the  two  methods  by  which  a  line 
given    in  perspective  may  be  divided   up  in  ^he  Method 
any   given   proportion  has    already   been   de-  ofi^^agonais. 
scribed.     It  was   shown   that   this,   though   called   the 
Method  of  Diagonals,  finally  leads  to  the  division   of 


54  MODERN  PERSPECTIVE. 

such  a  line  by  means  of  a  triangle,  one  side  of  which  is 
formed  by  the  line  to  be  divided,  and  one  side  by  an 
auxiliary  line,  drawn  parallel  to  the  plane  of  the  picture 
in  any  convenient  direction  and  divided  in  the  given 
proportion.  The  points  of  division  are  transferred  from 
this  auxiliary  line  first  to  the  third  side  of  the  triangle, 
by  lines  parallel  to  the  perspective  line,  and  directed  to 
its  vanishing  point ;  and  then  to  the  perspective  line  by 
lines  actually  parallel  to  the  auxiliary. 

69.  Both  these  steps  are  obvious  and  simple  applica- 
tions of  the  proposition  that  lines  drawn  parallel  to  one 
side  of  a  triangle  divide  the  other  two  sides  proportionally. 
But  it  does  not  yet  appear  what  is  the  real  direction  of 
this  third  side,  nor  in  what  plane  the  triangle  really  lies  ; 
that  is  to  say,  the  vanishing  point  of  this  line  and  the 
lioi'izon  of  tills  ])lane  are  not  yet  determined. 

70.  The  other  method  of  dividing  perspective  lines, 
™   „  .  .    called,  par  excellence,  the  Method  of  Triangles, 

The  Method  '  ^  '  &       > 

of  Triangles,  ^g  ^  niorc  dircct  application  of  the  same  prin- 
ciple. The  auxiliary  line,  as  before,  is  drawn  parallel  to 
the  plane  of  the  picture ;  but  the  points  by  which  it  is 
divided  are  now  transferred  directly  to  the  perspective 
line  by  lines  drawn  parallel  to  the  third  side  of  the  tri- 
angle. Plate  III.  is  devoted  to  the  illustration  of  this 
method.  Fig.  7,  a  and  h,  shows  the  difference 
between  this  method  and  the  preceding.  In 
each  of  the  triangles  here  shown,  the  base  is  divided  pro- 
portionally to  the  parts  set  off  on  the  left-hand  side.  But 
in  the  upper  ones  the  division  is  effected  by  the  Method 
of  Diagonals,  as  in  Fig.  3,  /;   in  the  lower  ones   the 


Fig.  i,a,h. 


THE   LIVISION    OF   LINES,   ETC.  55 

same  result  is  reached,  more  directly  and  simply,  by  the 
Method  of  Triangles. 

71.  This  application  of  the  principle  in  question, 
however,  though  more  direct  and  simple,  is  in  one  re- 
spect less  easy  of  adaptation  to  lines  given  in  perspec- 
tive. For  the  two  systems  of  parallel  lines  employed  in 
the  Method  of  Diagonals  may  be  drawn  without  diffi- 
culty, the  first  having  the  same  vanishing  point  as  the 
line  to  be  divided,  and  tlie  second  being  actually  parallel 
to  the  auxiliary  line,  since  that  line  is  parallel  to  the 
picture.  But  in  the  Method  of  Triangles,  the  lines  by 
which  the  points  are  transferred  are  parallel  to  the 
third  side  of  the  triangle,  whose  vanishing  point  is  not 
known.  It  is  accordingly  necessary  first  to  find  the 
vanishing  point  of  this  line. 

72.  This  may  be  done  at  once,  when,  as  in  the  plan 
of  the  eaves,  at  the  top  of  the  plate,  the  plane  Lhieof^Pro- 
in  which  the  auxiliary  triangle  lies  is  known ;  Mealurel 
that  is  to  say,  wlien  its  horizon  has  been  already  ascer- 
tained.    The  auxiliary  line  here  lies  in  the  horizontal 
plane,  and  the  given  line  lies  in  the  same  Horizontal 
plane  ;  the  whole  triangle  is  accordingly  in  the  ^"^^' 
horizontal  plane,  and  all  its  lines  have  their  vanishing 
points  in  the  Horizon,  —  the  given  line  at  V^,  the  auxil- 
iary line  at  an  infinite  distance,  and  the  third  side  of  the 
triangle  at  Vj.    This  point  is  ascertained  simply  by  pro- 
longing this  side  until  it  reaches  the  Horizon  (13  I.).    If 
now  it  is  desired  to  find  the  position  of  the  ten  brackets 
that  support  the  eaves,  it  is  easy  to  lay  off  on  the  auxiliary 
line  nine  equal  divisions,  and  to  complete  the  triangle  : 


56  MODERN  PERSPECTIVE. 

by  drawing  lines  parallel  to  the  third  side,  the  distances 
set  off  on  the  auxiliary  are  at  once  transferred  to  the  per- 
spective line,  the  lines  converging  to  Vj.  This  auxiliary 
vanishing  point  is  called  the  vanishing  point  of  propor- 
tional measures,  or  simply  the  point  of  measures.  The 
auxiliary  line  is  called  the  line  of  proportional  measures, 
or  simply  the  line  of  measures.  These  must  not  be  con- 
founded with  the  point  and  line  of  eqiial  measures  de- 
scribed in  the  next  chapter  (98,  99,  125). 

73.  It  makes  no  difference,  of  course,  at  which  end  of 
the  perspective  line  the  line  of  proportional  measures  is 
drawn,  so  that  it  is  parallel  to  the  picture.  The  relative 
position  of  the  doors,  windows,  etc.,  in  the  lower  plan, 
and  of  the  posts  of  the  shed,  are  laid  off  on  lines 
of  measures  drawn  from  their  further  ends,  and  the 
points  of  division  transferred  to  the  lines  of  the  plan 
by  means  of  the  points  of  measures  V2  and  V3,  both  of 
which,  of  course,  are  also  on  the  Horizon.  But  it  is  ob- 
viously conducive  to  precision  to  have  the  line  of  mea- 
sures touch  the  nearer  end  of  the  line  to  be  divided, 
since,  in  general,  converging  lines  give  more  accurate  re- 
sults than  do  lines  of  divergence. 

Neither,  of  course,  does  the  size  of  the  proportional 

parts  laid  off  upon  the  line  of  measures  affect  the  result. 

In  Fig.  7,  c,  the  base  of  the  triangle  is  divided 

Fig.7,c.        .  ^ 

into  the  same  four  equal  parts,  whether  the 
parts  taken  on  the  adjacent  side  are  large  or  small. 
Any  convenient  scale  may  be  used  ;  but  that  scale  will 
in  general  be  found  most  convenient  which  makes  the 
line  of  measures  about  as  long  as  the  perspective  line  to 


THE   DIVISION    OF   LINES,   ETC.  67 

be  divided,  and   which  brings  the  point   of   measures 
within  easy  reach. 

74.  In  the  same  way  a  line  lying  in  a  vertical  plane 
may  be  divided  by  means  of  a  vertical  line  of  vertical 
measures ;  the  point  of  measures  or  vanishing  ^^^* 
point  of  the  third  side  of  the  triangle  and  of  the  lines 
drawn  parallel  to  it  being  now  in  the  horizon  of  the  ver- 
tical plane.  If  a  line  lies  at  the  intersection  of  two 
planes,  it  is  a  mere  matter  of  convenience  whether  the 
line  of  measures  is  taken  in  one  plane  or  the  other,  or  in 
which  horizon  the  point  of  measures  is  taken. 

Thus  the  seven  parts  into  which  the  length  of  the  barn 
in  Plate  HI.  is  divided  may  be  taken  either  on  a  hori- 
zontal or  on  a  vertical  line;  that  is  to  say,  upon  a  line 
of  measures  parallel  to  the  horizon  of  either  plane.  The 
points  at  the  bottom  of  the  wall,  on  the  left-hand  side, 
which  determine  the  position  of  the  doors  and  windows, 
may  be  got  either  by  means  of  a  horizontal  line  of 
measures,  as  shown,  with  its  point  of  measures  on  the 
Horizon,  at  V4,  or  by  a  vertical  line  of  measures,  namely, 
the  corner' of  the  barn,  on  which  the  same  proportional 
parts  are  laid  off,  with  its  point  of  measures  on  tlie  hori- 
zon of  the  plane  LZ  at  V5.  Here  the  first  triangle  lies 
in  the  horizontal  plane,  and  the  second  in  the  vertical 
plane,  the  first  on  the  ground  and  the  second  in  the 
side  of  the  barn,  as  they  seem  to. 

75.  If  a  line  lies  in  a  plane  inclined  to  the  horizontal 
plane,  as  each  inclined  line  of  the  gable-ends  indjned 
of  the  barn  lies  in  the  plane  of  its  roof,  a  si  mi-  ^"®^' 


58  MODERN    PERSPECTIVE. 

lar  procedure  may  be  followed.  A  line  of  measures  may 
be  taken  in  that  plane,  touching  the  given  line  at  one 
end  and  parallel  to  the  picture,  the  point  of  measures 
being  now  in  the  horizon  of  the  plane  of  the  roof. 

76.  And  as  in  the  horizontal  plane  a  line  parallel  to 
the  picture  is  horizontal,  and  in  vertical  planes  vertical, 
—  that  is  to  say,  in  each  case  parallel  to  the  horizon  of 
the  plane  it  lies  in,  —  so  in  the  case  of  an  inclined  plane, 
a  line  lying  in  it  parallel  to  the  picture  is  parallel  to  the 
horizon  of  the  system  to  which  the  plane  belongs;  for 
though  its  vanishing  point  is  somewhere  in  that  horizon 
(13, 1.),  it  is  at  an  infinite  distance  (20).  A  line  directed 
to  that  point  is  accordingly  parallel  to  the  horizon. 

77.  The  perspective  of  an  inclined  line  can  then  be 
divided  in  any  required  proportion,  as  easily  as  that  of 
a  horizontal  or  vertical  one,  by  drawing  through  one  end 
of  it  a  line  of  proportional  measures  parallel  to  the  hori- 
zon of  the  inclined  plane  in  which  it  lies,  and  taking  the 
point  of  measures  on  that  horizon. 

Thus  in  the  plate  the  position  of  the  brackets  or  pur- 
lins on  the  gable  of  the  barn  is  found  by  dividing  each 
slope  into  six  parts,  by  means  of  a  line  of  measures 
drawn  parallel  to  H  LM,  the  horizon  of  the  roof  in  ques- 
tion; and  as  the  sloping  lines  of  the  gable  lie  in  a  vertical 
plane  R  Z,  parallel  to  the  side  of  the  barn,  as  w^ell  as  in 
the  plane  of  the  roof,  it  follows  that  the  position  of  the  six 
brackets  can  be  found  either  by  laying  off  equal  parts,  on 
vertical  lines,  with  points  of  measures  on  the  horizon  of 
R  Z,  at  Vg  and  Y7,  or  by  laying  off  equal  parts  upon  lines 
of  measures  parallel  to  the  horizons  of  the  planes  of  the 


THE    DIVISION    OF    LINES,    ETC.  59 

roofs,  that  is  to  say,  parallel  to  H  L  M  for  the  left-hand 
slope,  and  to  H  L  M'  for  the  other,  with  points  of  propor- 
tional measures  at  Vg  in  HLM,  and  Vg  in  HLM',  re- 
spectively. 

In  the  former  case  the  triangles  lie  in  the  plane  of 
the  gable-end ;  in  the  latter,  each  lies  in  the  plane 
of  its  own  roof. 

78.  It  follows  from  the  above,  as  has  already  been 
shown,  that  if  any  ol)ject  bounded  by  plane   Sections 

P^^rallel  to 

surfaces  be  cut  through  by  a  plane  parallel  to  the  picture, 
the  plane  of  the  picture,  the  line  of  intersection  on  each 
face  will  be  parallel  to  the  horizon  of  the  plane  in  which 
it  lies  (39).  This  is  exemplified  in  the  plate,  where  the 
dotted  line,  A  A,  running  along  the  ground  and  over  the 
barn,  follows  this  law.  If  the  front  corner  of  the  build- 
ing were  sliced  off  parallel  with  the  picture,  this  would 
be  the  line  of  the  cut.  The  same  thing  is  exemplified 
on  the  front  corner  of  the  other  building. 

We  shall  find  use  for  this  by  and  bye,  when  we  come 
to  the  perspective  of  shadows. 

79.  Finally,  just  as  in   the  Method  of  Diagonals  we 

found  at  last  that  the  auxiliary  line,  or  line  of  J^andom 

J         '  hues  of  pro- 

measures,  may  be  taken  in  any  direction,  at  measures, 
random,  so  here  the  same  thing  is  true.  For  here  too 
any  line,  drawn  in  any  direction  at  random,  from  either 
end  of  a  perspective  line,  may  be  regarded  as  the  per- 
spective of  a  line  of  measures,  parallel  to  the  picture 
and  drawn  parallel  to  the  horizon  of  the  plane  in  which 
it  lies.     This  horizon,  then,  will  be  parallel  to  it;  and 


60  MODERN    PERSPECTIVE. 

since  the  plane  contains  the  perspective  line,  its  horizon 
must  pass  through  the  vanishing  point  of  that  line ;  for 
the  horizon  of  a  plane  passes  through  the  vanishing 
points  of  all  the  lines  that  lie  in  it  (13  c);  if  then 
through  the-  vanishing  point  of  the  line  we  wish  to 
divide,  we  draw  a  line  parallel  to  the  assumed  line  of 
measures,  we  shall  have  the  horizon  of  a  plane  in  which 
they  both  lie;  and  upon  this  horizon  the  third  line  of 
the  triangle,  joining  the  other  end  of  the  perspective 
line  with  the  last  point  taken  on  the  line  of  proportional 
measures,  wull  have  its  vanishing  point.  This  point, 
the  point  of  measures,  can  be  found,  just  as  before,  by 
prolonging  the  third  line,  the  base  of  the  triangle,  till 
it  touches  it. 

80.  The  principle  that  the  line  of  measures  may  be 
drawn  at  random  in  any  direction,  the  corresponding 
point  of  measures  being  taken  on  a  line  or  horizon 
drawn  parallel  to  it  through  the  vanishing  point  of  the 
line  to  be  divided,  is  illustrated  in  the  division  into  five 
equal  parts  of  the  hip  of  the  roof  of  the  smaller  building 
in  the  middle  distance.  Here  the  line  of  measures  is 
drawn  arbitrarily  at  about  60°,  the  auxiliary  horizon  be- 
ing drawn  through  V^,  the  vanishing  point  of  the  hip,  and 
its  point  of  measures,  V^o  determined  on  that  horizon. 

The  triangle  here  seems  to  lie  in  the  plane  of  the 
roof,  but  in  fact  it  has  nothing  to  do  with  it. 

81.  Moreover,  since  the  only  characteristic  of  this 
auxiliary  horizon,  relatively  to  the  conditions  of  the 
problem,  is  this,  that  it  passes  through  the  vanishing 


THE    DIVISION    OF   LINES,   ETC.  61 

point  of  the  line  to  be  divided,  it  follows  that  any  line 
drawn  through  the  vanishing  point  of  a  given  line  may 
be  regarded  as  the  horizon  of  a  plane  in  which  the  given 
line  lies,  and  will  contain  the  point  of  measures  cor- 
responding to  a  line  of  measures  drawn  through  either 
end  of  the  given  line  parallel  to  it. 

82.  This  gives  us,  in  other  words,  this  singular  prop- 
osition :  — 

Of  any  two  perspective  lines  having  the  same  vanish- 
inu:  point,  one  may  be  taken  as  the  horizon  of  a 

^  ^  *^  _  Perspective 

plane  passing  throudi  the  other  ;  and  if  a  third   ^'''*'^  ".^?*^ 

^  -t  o  o  y  as  auxiliary 

line  be  drawn  parallel  to  the  first,  and  touch-  ^°''^o°^- 
ing  one  end  of  the  second,  any  parts  taken  upon  this 
third  line  may  be  transferred  to  the  second  in  their 
true  proportions  by  means  of  a  point  of  measures  taken 
upon  the  first.  See  Gwilt's  Encyclopaedia  of  Architec- 
ture, §  2457. 

83.  The  position  of  the  vertical  bars  of  the  cresting 
upon  the  ridge  of  the  gate-house  on  the  left  is  deter- 
mined in  this  way,  five  equal  parts  being  laid  off  upon  a 
line  drawn  from  the  further  end  of  the  ridge  parallel 
to  the  eaves  of  the  roof,  as  a  line  of  measures,  and  the 
point  of  measures,  Y^,  taken  on  the  eaves. 

The  way  in  which  the  position  of  the  vertical  bars  of 
the  gate  below  is  determined  also  illustrates  this  prop- 
osition. A  line  touching  the  top  of  the  gate  is  drawn 
parallel  to  the  ridge-pole,  which  has  the  same  vanishing 
point,  Y^  Equidistant  points  are  taken  on  this  line, 
and  transferred  to  the  top  of  the  gate  by  a  point  of  mea- 
sures, Yi2,  taken  on  the  ridge-pole.     This  reduces  the 


62  MODERN   PEKSPECTIVE. 

labor  of  dividing  up  a  given  perspective  line  in  any  re- 
quired proportion  to  almost  nothing. 

84.  Here,  as  in  the  corresponding  case  in  the  pre- 
vious chapter,  care  is  to  be  taken  not  to  fancy  that  the 
line  of  measures,  and  the  triangle  determined  by  it,  really 
lie  in  the  plane  they  seem  to  lie  in. 

In  this  last  case  the  triangle  lies  in  an  imaginary  in- 
clined plane,  and  is  no  more  vertical,  as  it  seems  to  be, 
than  the  point  of  measures  is  on  the  ridge,  as  it  seems 
to  be :  it  is  really  in  the  infinitely  distant  horizon  which 
the  ridge  covers  and  coincides  with. 

Plate  III.  also  furnishes  illustrations  of  two  points  of 
general  interest. 

85.  The  first  of  these  is  the  use  of  the  point  V-^,  the 
vanishing  point  of  horizontal  lines  making  an  angle  of 
45°  with  the  principal  directions  E  and  L,  to  determine 
V^,  when  V^  is  given,  the  lines  that  slope  up  to  the  left 
being  supposed  to  make  the  same  angle  with  the  ground 
as  those  that  slope  up  to  the  right.  If  these  inclinations 
are  equal,  the  inclination  of  the  planes  EN  and  LM 
will  be  equal,  as  in  the  case  of  these  roofs  ;  their  lines  of 
intersection,  P,  will  lie  in  vertical  planes,  making  45°  with 
the  principal  vertical  planes  ;  their  horizon  will  be  a  ver- 
tical line  passing  through  V-^,  as  shown  in  the  previous 
chapter  (42) ;  and  Y^  will  be  at  the  intersection  of  this 
horizon  with  H  L  M.  If  now  H  E  N  be  drawn  through 
V^  and  N^,  V^  will  be  found  at  its  intersection  with 
HLZ,  and  V^'  will  be  at  an  equal  distance  below. 

The  point  V^\  which  determines  the  direction  of  the 


THE   DIVISION    OF    LINES,   ETC.  63 

line  Pi,  at  the  intersection  of  the  two  banks  in  the  fur- 
ther corner  of  the  barnyard,  is  found  in  like  manner. 

86.  The  second  point  is  illustrated  by  Fig.  8,  which 
shows  how  the  true  direction  of  the  lines  Q  or  Q',  whose 
vanishing  points  are  at  the  distant  intersec-  Fig.  8. 
tion  of  the  nearly  parallel  horizons  HEN  and  H  L  M', 
or  H  R  N'  and  H  L  M,  may  be  obtained  by  means  of 
two  similar  triangles,  the  common  device  for  Fig.  a 
directing  a  third  line  to  the  intersection  of  two  given 
lines,  as  shown  in  Fig.  9. 

This  is  applied  in  the  plate,  Fig.  10,  to  find  the  true 
direction  of  the  left-hand  line  of  the  hipped  Fig.  lo. 
roof,  just  below  the  point  V^, 


CHAPTEE  V. 

ON    THE    EXACT   DETERMINATION    OF   THE   DIRECTION   AND 
MAGNITUDE   OF   PERSPECTIVE   LINES. 

THE  first  two  chapters  were  given  to  a  general 
observation  of  the  phenomena  of  perspective, 
in  nature  and  in  drawings,  and  the  last  two  to  an 
explanation  of  the  practical  making  of  such  draw- 
ings, certain  data  being  assumed.  It  was  assumed 
that  tlie  position  of  the  principal  vanishing  points, 
giving  the  direction  of  the  principal  lines,  had  been 
already  determined,  with  more  or  less  accuracy,  by 
the  eye  or  by  the  judgment,  and  that  their  length 
had  also  been  fixed  in  the  same  way.  The  discus- 
sion showed  how  the  position  of  other  vanishing 
points  and  the  length  and  direction  of  other  lines 
could  then  be  determined,  and  how  any  of  the 
lines  thus  drawn  could  be  divided  up  in  any  de- 
sired manner,  that  is  to  say,  in  any  given  propor- 
tion. 

It  is  necessary,  in  order  to  conclude  this  part  of  the 
subject,  to  show  how  these  data  may  be  more  precisely 
determined.  It  is  necessary  to  show  how,  when  the 
real  direction  of  lines  is  exactly  known,  their  vanishing 
points  may  be  fixed  with  precision,  and  how,  when  their 


PERSPECTIVE  LINES.  65 

real  length  is  known,  the  exact  length  and  position  of 
their  perspective  representations  may  be  determined. 
The  position  of  the  object  to  be  drawn,  the  position  of 
the  picture,  and  the  position  of  the  spectator's  eye, 
must,  of  course,  be  known  also. 

Plate  IV.  shows  how  these  questions  are  answered : 
Fig.  11  showing  in  plan,  and  upon  a  reduced  piateiv. 
scale,  the  position  of  the  spectator  at  the  sta-  ^'s-  lo- 
tion point  S ;  that  of  the  picture  at  pp,  which  shows 
the  plane  of  the  picture  edgewise  as  it  would  appear 
looking  down  upon  it ;  and  that  of  the  object  to  be  rep- 
resented at  A.  Two  elevations  of  this  object,  which  is 
a  small  house,  showing  its  vertical  dimensions,  are 
given  alongside ;  the  plan  and  the  two  elevations 
together  giving  exact  information  as  to  the  magni- 
tude and  direction  of  the  lines  defining  it.  The  pic- 
ture itself  is  shown  betw^een  the  plan  of  the  house 
and  its  own  plan,  just  as  if  the  plane  of  the  picture, 
P2J,  had  been  revolved  backward  into  the  plane  of  the 
paper. 

The  object  is  here  represented  as  being  about  six 
times  as  far  from  the  station  point  as  the  picture 
is,  the  picture  being  about  eighty  feet  from  the  sta- 
tion point,  and  the  nearest  corner  of  the  house  about 
thirteen  feet.  It  is  this  relation  that  obviously  de- 
termines the  scale  of  its  perspective  representation, 
which  would  be  greater  if  the  picture  were  farther 
from  the  station  point,  or  the  object  nearer,  and  vice 
versa.  But  we  shall  come  to  the  question  of  scale 
presently  (94). 

5 


66  MODERN   PERSPECTIVE. 

87.  The  first  question  is  that  of  the  direction  to  be 
given  to  the  various  perspective  lines ;  we  must  de- 
The  Direc-     temiine  the  vanishing  points  of  these  various 

tions  of  Hori- 
zontal Lines,  sjstems  of  liues,  horizontal  and  inclined.  Ver- 
tical lines,  and  all  other  lines  parallel  to  the  picture,  will 
of  course  have  their  perspectives  parallel  to  themselves 
(20).  The  horizontal  lines  belong  to  three  systems,  the 
directions  of  which  are  indicated  in  the  plan  of  the  little 
house  as  E  and  L,  going  off  to  the  right  and  to  the  left, 
at  right  angles  to  each  other ;  and  X,  dividing  the  angle 
between  them,  and  making  an  angle  of  45°  with  each. 
If  now  the  spectator,  standing  at  S-,  looks  in  a  direction 
parallel  to  E,  he  will  see  the  vanishing  point  of  that  sys- 
tem of  lines  directly  before  his  eye  ;  tliat  element  of  the 
system  which  passes  through  S  is  in  fact  seen  endwise, 
appearing  as  a  point  covering  and  coinciding  with  the 
vanishing  point  of  the  system  of  right-liand  horizontal 
lines,  which  is  in  the  infinitely  distant  horizon  (6).  The 
perspective  of  this  vanishing  point,  V^,  in  the  plane  of 
the  picture,  will  be  found  exactly  where  this  element 
pierces  the  picture,  that  is,  where  it  crosses  the  line 
pp,  V^  and  V^  can  of  course  be  found  in  the  same 
way  ;  and  the  centre  of  the  picture,  V^,  the  point  nearest 
the  station  point  and  at  the  other  extremity  of  the  axis 
S  Y°,  is  easily  determined  at  the  same  time.  Since  E  and 
L  are  at  right  angles,  the  triangle  V^  S  V^  is  a  right- 
angle  triangle,  and  S  lies  on  the  circumference  of  a  semi- 
circle of  which  Y^  and  V^  give  tlie  diameter.  In  the 
picture  itself,  just  above,  these  points  of  course  appear 
on  the  Horizon ;  for  since  these  lines  are  all  horizontal. 


PERSPECTIVE    LINES.  67 

their  vanishing  points  lie  in  the  horizon  of  the  horizontal 
system  of  planes  (13,  I.).  H  R  Z  and  H  L  Z,  the  hori- 
zons of  vertical  planes  parallel  to  the  sides  of  the  build- 
ing, can  now  be  drawn,  as  usual,  through  V^  and  V^; 
and  H  P  P',  the  horizon  of  the  diagonal  planes,  through 
V-^.  The  vanishing  points  of  any  of  the  other  horizon- 
tal lines,  and  the  horizons  of  any  other  vertical  planes 
could  of  course  be  found  in  a  similar  manner.  A  line 
perpendicular  to  the  plane  of  the  picture,  and  accord- 
ingly parallel  to  the  Axis,  would  of  course  have  its  van- 
ishing point  at  V*^. 

88.  It  only  remains  to  find  the  vanishing  points  of  the 
inclined  lines  M  and  M',  N  and  N',and  thence  TheDirec- 

.  tions  of  In- 

as  berore,  the  horizons  oi  the  roofs,  and  the  van-  cUuedLiues. 
ishing  points,  P  and  P',  of  their  hips  and  valleys.  This  is 
easily  done  by  the  aid  of  the  elevations,  which  show  the 
real  inclination  of  these  roofs  and  gables  to  be  60°  for 
the  lower  slope,  and  30°  for  the  upper.  If  the  spectator 
at  S,  then,  while  looking  at  V^  in  the  direction  R,  should 
raise  his  eyes  at  an  angle  of  30°,  he  would  see  Y^^,  the  van- 
ishing point  of  the  upper  slope  of  the  gable,  directly  be- 
fore him,  the  triangle  V^  V^  S  being  right-angled  at  V^. 
If  now  this  triangle  were  revolved  about  the  vertical 
side  V^  V^,  so  as  to  bring  the  station  point,  S,  into  the 
plane  of  the  picture  at  D^,  it  would  appear  in  the 
picture  above  as  the  triangle  V^^  V^  D^,  the  angle  at  D^ 
being  30°. 

Fig.    13  gives  a  perspective   view   of  the   plane   of 
the   picture  pp,  with  the  eye  at  the  station 
point,  S,  in  front  of  it ;  the  triangle  in  question 


6S  MODERN  PERSPECTIVE. 

is  shown  both  in  its  original  position  and  also  as  it  ap- 
pears when  swung  round  into  the  plane  of  the  picture. 

89.  It  follows  from  this  that  if  from  V^  the  distance 
V^  S  is  laid  off  along  the  Horizon,  we  obtain  the  point 
D^ ;  and  if  from  this  point  we  draw  a  line  at  an  angle 
of  30°,  we  shall  obtain  V^  at  its  intersection  with 
HRZ.  In  the  same  way  D^  and  V^  may  be  obtained 
by  setting  off  the  distance  of  V^  S  along  the  Horizon 
from  V^,  and  drawing  the  line  D^  V^,  also  at  an  angle 
of  30°. 

The  points  D^  and  D^  are  called  the  right-hand  and 
Points  of  Dis-  left-hand  points  of  distance ;  they  show  the 
*^°^®"  distance  of  the  station  point  from  the  right- 

hand  and  left-hand  vanishing  points.  It  is  to  be  ob- 
served that  D^  is  found  on  the  left  and  D^  on  the  right 
of  V^. 

In  the  same  way  the  vanishing  point  of  every  other 
horizontal  line,  as,  for  example,  of  X,  has  its  correspond- 
ing point  of  distance,  found  by  setting  off  along  the  hori- 
zon its  distance  from  S.  Thus  we  have  V^"^!)-^  equal 
to  Y^^S. 

90.  V^'  and  V^*'  will  of  course  be  seen  as  far  be- 
low the  Horizon  as  Y^  and  Y^  are  above  it,  and  Y^ 
and  Y^'  will  be  at  the  intersection  of  the  horizons  of 
the  inclined  phanes  R  N  and  L  M,  K  N'  and  L  M',  as 
before. 

Fig.  13  gives  further  illustration  of  most  of  tliese 
points.  To  prevent  a  confusion  of  lines,  the  centre,  Y^^ 
is  taken  on  the  left-hand  side  of  the  picture  instead  of 
on  the  right-hand  side,  as  in  Fig.  11. 


PERSPECTIVE   LINES.  69 

91.  By  a  reverse  process,  when  the  vanishing  point 
of  inclined  lines  is  known,  their  real  inclina-   to  find  the 
tion  can  be  discovered  by  drawing  a  line  from   do"n  Jnn-"*" 
this  vanishing  point  to  the  point  of  distance  of  ^hose  per- 

sjiectives  are 

the   horizontal  line  beneath  them.     Thus   in  ^'^''''■ 
the  figure  a  line  from  V^  to  D^  would  give  the  angle 
yp  j)x  yx  ^vhich  is  the  true  slope  of  the  line  of  intersec- 
tion of  the  upper  roofs. 

92.  If  the  lines  M  and  N  have  different  inclinations, 
the  point  V^  will  of  course  not  come  over  V-'^,  and  the 
distance  must  be  measured  from  the  point  of  the  hori- 
zon that  it  does  come  over. 

The  vanishing  points  of  the  steeper  slopes  of  the 
lower  roofs  are  found  in  like  manner  at  Y^\  V^^,  etc. 

93.  The  exact  direction  of  perspective  lines  being 
thus  determined,  since  the  position  of  their 

^  The  length 

vanishing  points  is  thus  exactly  fixed,  it  now  afkUo't^hJ' 
only  remains  to  determine  their  length,  and  p''^^"'"®- 
the  position  of  some  one  point  in  each.  Tor  such  lines 
as  are  parallel  to  the  picture  this  is  easy,  for  every 
such  line  may  be  considered  to  lie  in  a  plane  parallel  to 
the  picture,  the  centre  of  which,  or  point  in  the  plane 
nearest  the  eye,  will  have  its  perspective  at  V^,  the  centre 
of  tlie  picture ;  the  perspective  of  the  line  in  question 
will  be  parallel  to  the  line  itself,  and  its  length  and  its  dis- 
tance from  the  centre,  Y*^,  and  all  other  lengths  and  dis- 
tances taken  in  that  plane,  will  be  less,  as  we  have  just 
seen  (86),  in  proportion  as  the  distance  of  the  plane  from 
the  plane  of  the  picture  is  greater.     If,  as  in  Fig.  11, 


70  MODERN   PERSPECTIVE. 

the  plane  m  m  is  six  times  as  far  from  the  spectator  at 
S  as  the  picture  p  p,  all  lines  in  m  m  will  be 

Fig.  11.  .  . 

drawn  at  one  sixth  of  their  original  size,  and  be 
at  one  sixth  their  distance  from  the  centre.  The  front 
corner  of  the  house,  for  instance,  whicli  lies  in  the  plane 
m  m,  is  so  drawn. 

This  imaginary  plane  7n  m,  which  is  generally  drawn 
The  Plane  of  througli  tlic  ucarcst  part  of  any  object  to  be 
Measures.  represented,  is  called  a  ^j/a?ic  of  measures, 
and,  like  the  picture,  is  defined  in  position  by  the 
length  and  position  of  its  axis,  which  coincides  with 
that  of  the  picture,  but  is  generally  a  great  many 
times  as  long. 

Its  relative  distance  behind  the  plane  of  the  picture 
is  commonly,  of  course,  much  greater  tlian  that  shown 
in  the  figure,  in  which,  for  perspicuity's  sake,  the  pic- 
ture is  represented  as  being  about  eleven  feet  across  and 
about  thirteen  feet  from  the  spectator.  The  picture  is 
commonly  set  only  a  few  feet  off,  wdiile  the  object  rep- 
resented is  often  a  hundred  times  as  many. 

Sometimes,  for  convenience,  two  planes  of  measures 
are  employed  at  different  distances,  lines  in  the  further 
one  being  drawn  to  a  smaller  scale  tlian  in  the  nearer  one. 

94.  It  follows  from  what  has  been  said  that  any  line 
Lines  in  tiae    drawu  iu  a  plauc  of  measures  in  anv  direc- 

Planeof  ^  _  ^  _  ... 

Measures.  i\QYi,  horizoutal,  vcrtical,  or  inclined,  is  also  par- 
allel to  the  picture,  and  that  its  perspective  will  be  parallel 
and  proportional  to  it,  but  on  a  smaller  scale.  This  scale 
depends  on  the  relative  distance  of  the  picture  plane 
and  the  plane  of  measures  from  the  eye,  or  station  point. 


PERSPECTIVE   LINES.  71 

If  the  latter  is  twice  or  ten  times  as  far  away,  lines  drawn 
upon  it  will  be  presented  in  the  picture  one  half  or  one 
tenth  full  size.  All  lines  in  a  plane  of  'measures  have 
their  perspectives  drawn  to  the  same  scale. 

It  is  common,  in  the  case  of  large  objects,  such  as 
buildings,  to  set  the  picture  at  just  -^\,  jl^,  geaie.ofthe 
or  -^^2  of  the  distance  of  the  plane  of  mea-  '^^"^'^'^s- 
sures,  i.  e.,  of  the  object.  Lines  in  the  plane  of  mea- 
sures are  then  represented  ^q,  y^^,  or  j^^  ^^^^^  size, 
etc. ;  that  is  to  say,  on  a  scale  of  ^,  ^2 ,  or  Jg  of  an  inch 
to  the  foot. 

The  centre  of  the  plane  of  measures  coincides  in  per- 
spective with  the  point  V^,  the  centre  of  the  picture  (93). 
The  perspective  of  any  other  point  in  the  plane  of  mea- 
sures may  be  found  by  laying  off  its  distance,  according  to 
the  scale,  in  its  real  direction,  in  the  plane  of  the  pic- 
ture. 

95.  Fig.  12,  in  which  the  various  vanishing  points  and 
traces,  and  the  points  of  distance  D^  and  D^, 

Fig.  12. 

are  determined  as  in  the  previous  figure,  illus- 
trates this  practice.  The  picture  is  supposed  to  be  about 
six  inches  from  the  eye,  as  in  the  case  of  our  previous 
illustrations.  This  is  less  than  is  desirable,  but  is  as 
much  as  the  scale  of  these  illustrations  permits.  The 
front  corner  of  the  building,  through  which  the  plane  of 
actual  measures  is  taken,  is  supposed  to  be  a  hundred 
and  ninety-two  times  as  far  away,  that  is  to  say,  nine- 
ty-six feet  from  the  spectator,  or  ninety-five  feet  and 
a  half  behind  the  picture,  the  scale  of  the  perspec- 
tive of  that  corner  being  one  sixteenth  of  an  inch  to 


72  MODERN   PERSPECTIVE. 

a  foot.  This  corner,  being  eight  feet  high,  is  drawn 
half  an  inch  high.  All  other  lines  in  the  plane  of 
measures  are  drawn  to  the  same  scale,  which  is  indeed 
the  same  scale  as  that  to  which  the  plan  of  the  build- 
ing is  drawn  in  Fig.  11,  and  the  elevations  alongside. 
Dimensions  can  then  be  transferred  directly  from  these 
drawings  to  Fig.  12,  so  long  as  the  lines  to  which 
they  apply  lie  in  the  plane  of  measures.  Such  a 
line  is  the  line  v  v,  the  line  in  which  the  planes  of 
the  front  and  end  walls  of  the  house  intersect  the  plane 
of  measures.  Any  such  vertical  line,  lying  in  the  plane 
of  measures,  is  called  a  line  of  vertical  measures.  By 
LineofVer-    settinsj  off  uDou  tliis  Huc  tlic  hciohts  of  anv 

ticalMea-  O  I  &  J 


part  of  the  building,  by  scale,  they  can  be 
transferred  to  their  proper  places  by  horizontal  lines 
directed  to  the  vanishing  points.  The  height  of  the 
gable  in  the  figure  is  determined  in  this  way.  The 
line  m  m,  also,  in  which  the  plane  of  measures  intersects 
the  horizontal  plane  on  which  the  perspective  plan  is 
taken,  is  such  a  line.  Its  perspective,  shown  at  g  I,  is 
LineofHori-  callcd  the  Grouud  Line,  or  line  of  horizontal 

zontal  Mea-  .  i        i     •  J        rv- 

sures.  measures,  and  any  dimensions  can  be  laid  on 

upon  it  at  the  same  scale  as  upon  the  perspective  of 
the  vertical  line ;  as  it  is  parallel  to  the  picture,  the 
divisions  of  its  perspective  are  proportional  to  those  of 
the  line  itself 

By  prolonging  the  planes  of  the  other  walls  until 
they  intersect  the  plane  of  measures,  additional  lines 
of  vertical  measures  are  obtained,  such  as  v'  v',  in  the 
plane  of  the  further  gable.     In  the  same  way  every 


PERSPECTIVE   LINES.  73 

horizontal  plane  gives  a  line  of  horizontal  meaGures,  as 
is  shown  in  the  case  of  the  two  perspective  plans  below. 

96.  For  very  small  objects  the  plane  of  measures,  and 
with  it  the  object  itself,  is  brought  nearer,  and  may  even 
coincide  with  the  plane  of  tlie  picture.  In  The  plane  of 
this   case   lines   lying   in   it   are   drawn   full  thepianrof 

measures. 

size. 

Sometimes,  instead  of  taking  the  object  of  its  real  size 
at  its  real  distance,  we  suppose  a  miniature  of  The  model  or 

miniature  ob- 

the  object  to  be  set  up  near  at  hand,  of  any  con-  ject. 
venient  scale.    In  this  case  also  the  object  may  be  sup- 
posed to  be  close  to  the  plane  of  the  picture,  and  the  plane 
of  the  picture  to  coincide  with  the  plane  of  measures. 

This  is  illustrated  in  Fig.  11,  in  which  a  small  plan 
of  the  house  is  drawn  in  contact  with  the  plane  of  the 
picture,  pp,  just  as  a  large  plan,  representing  full  size, 
is  drawn  in  contact  with  the  plane  of  measures  mm 
above,  the  whole  being  drawn  at  a  scale  of  sixteen 
feet  to  the  inch.  Or  w^e  may  regard  Fig.  11  as  drawn 
on  the  same  scale  as  Fig.  12,  that  is  to  say,  full  size,  con- 
sidering the  plane  m  m  to  be  the  plane  of  the  picture, 
six  inches  from  the  eye  at  S,  with  a  miniature  of  tlie 
building,  a  model  made  to  the  scale  of  a  sixteenth  of  an 
inch  to  the  foot,  just  behind  it.  In  this  case  Fig.  12 
represents  the  picture  drawn  on  the  plane  m  m,  just  as 
the  little  picture  in  Fig.  11  represents  that  drawn  on 
plane  pp. 

97.  Having  thus  the  means  of  drawing  in  any  hori- 


74  MODERN   PERSPECTIVE. 

zontal  or  vertical  plane  a  line,  lying  in  a  plane  of  mea- 
The  length     suves,  upon  wliicli  climensions  can  be  laid  off 

of  other  lines    ,  ,  ,  ^      ,  n       .i  t 

parallel  to  the  Dj  scalc,  wc  liave  now  to  transfer  thesc  dimen- 

picture,  by  i  t  •         i 

scale.  sions  to  otlier  lines  m  the  same  plane. 

If  these  lines  also  are  parallel  to  the  picture  and  to 
the  plane  of  measures,  the  case  presents  no  difficulty. 
It  is  only  necessary  to  draw  parallel  lines  from  one  line 
to  the  other.  In  the  figure,  for  example,  the  heights 
laid  off  on  the  line  of  vertical  measures  vv,  at  the 
front  corner  are  transferred  to  the  other  corners  and  to 
other  vertical  lines  by  parallel  lines  directed  to  the  van- 
ishing points  V^  and  V^.  In  this  way  the  vertical  di- 
mensions of  every  part  of  an  object,  and  the  position  of 
its  horizontal  lines,  may  be  determined. 

The  length  of  any  other  lines  parallel  to  the  picture, 
horizontal,  vertical,  or  inclined,  may  be  obtained  in  a 
similar  way  from  lines  in  the  plane  of  measures  parallel 
to  them,  and  lying  at  the  intersection  of  that  plane 
with  the  planes  in  wliich  they  lie. 

98.  To  determine  the  length  of  the  horizontal  lines 
The  length  of  not  parallel  to  the  picture,  and  to  layoff  given 

horizontal  .  •  r  ^     t 

hnes  inclined  dimcnsious  upou  thc  pcrspcctivc  01  such  lines, 

to  the  picture 

by  scale.  yy^  qq^^^  cmploy  a  method  similar  to  the  Method 
of  Triangles  described  in  the  last  chapter.  By  that  meth- 
od we  laid  off  upon  such  lines  parts  proportional  to  parts 
taken  upon  a  line  of  2^ropo7^tional  measures.  We  now 
propose  to  lay  off  upon  such  perspective  lines  parts 
equal  to  parts  taken  upon  a  line  of  real  measures.  Any 
triangle  will  do  to  transfer  proportional  parts,  but  tc 


PERSPECTIVE   LINES.  75 

transfer  equal  parts  we  must  have  an  isosceles  triangle ; 
for  it  is  only  in  isosceles  triangles  that  the  parts  into 
which  the  adjacent  sides  are  divided  by  lines  drawn 
parallel  to  the  base  are  equal,  each  to  each. 

This  is  illustrated  by  Fig.  11,  in  which  the  line  mm^ 
at   the   top,  is  the   line   of  horizontal   mea- 
sures.    The   actual  dimensions    of  the   sides  equal  mea- 
sures. 

of  the  house  and  of  the  doors  and  windows 
are  laid  off  on  this  line,  and  connected  with  the  in- 
clined  lines   of  the   plan   by   means    of  lines   drawn 
parallel  to  the  base  of  an  isosceles  triangle. 

99.  It  is  plain  that  what  is  here  done  in  the  ortho- 
graphic plan  could  be  done  in  a  perspective  plan  if  we 
knew  in  what  direction  to  draw  these  parallel  lines ; 
that  is  to  say,  if  we  could  find  the  vanishing  point  of 
the  base  of  the  isosceles  triangle. 

And  this  is,  in  fact,  very  easy,  for  a  simple  inspection 
of  the  figure  shows  that  the  j^oint  of  distance  is  the 
auxiliary  vanishing  point  in  question.     If  the  The  Point  of 

Distance, 

spectator  atS  looks  in  the  direction  of  the  par-  again,  as  a 

*  ^  point  of  equal 

allel  lines  by  which  the  right-hand  line  E  is  ^''asures. 
divided,  he  will  see  D^  ;  and  in  like  manner  D^  is  the 
vanishing  point  of  the  parallels  by  which  distances  taken 
on  the  line  of  horizontal  measures  are  transferred  to  L. 

And  that  this  is  as  it  should  be  is  plain  from  a  fur- 
ther inspection  of  the  figure.  For  the  sides  of  the 
isosceles  triangles  at  the  top  are  by  construction  parallel 
to  the  sides  of  the  triangles  SY^  D«  and  S V^  Bh  These 
last  are  accordingly  isosceles  too,  and  their  two  long- 
sides  should  be  equal.     The  auxiliary  vanishing  points 


76  MODEKN   PERSPECTIVE. 

then,  should  be  just  as  far  from  the  vanishing  points  as 
these  last  are  from  the  station  point ;  as  the  points  of 
distance  are,  by  construction.  Y^  D^  was  originally 
taken  equal  to  V«  S,  and  Y^  D^  to  Y^  S  (89). 

100.  The  points  of  distance,  then,  are  the  vanishing 
points  of  the  parallel  lines  which  will  intercept  upon 
a  perspective  line  parts  equal  to  those  intercepted  upon 
its  line  of  measures. 

This  is  illustrated  in  Fig.  12,  where,  in  the  perspec- 
tive plan,  parts  laid  off  by  scale  on  the  ground  line,  or 
line  of  horizontal  measures,  are  transferred  in  their  true 
dimensions  to  the  perspective  lines  K  and  L.  In  this 
way  the  length  of  the  walls  and  the  position  of  the 
doors  and  windows  is  exactly  determined.  These  di- 
mensions being  already  shown  at  the  given  scale  in  the 
little  elevations,  it  suffices  to  transfer  them  directly  from 
those  drawings  to  the  ground  line  with  a  measuring  strip. 

In  this  way  a  complete  and  accurate  perspective  plan 
The  Method  ^^^^  easily  ])e  constructed  ;  the  length  of  all  hori- 
spective  ^^'  zontal  lines  and  the  position  of  all  vertical  lines 
will  then  be  known.  The  length  of  vertical 
lines,  which  gives  the  position  of  horizontal  ones,  is  easily 
obtained,  as  we  have  seen,  from  vertical  lines  of  measures. 

The  second  perspective  plan,  above  the  other,  gives 
the  plan  of  the  roof  and  dormers. 

101.  Fig.  11  affords  an  alternative  method  of  obtain- 
Fig.  11.      ing  the  horizontal  dimensions ;  tliat  is  to  say, 

The  Method    the  positioii  of  the  vertical  perspective  lines. 

of  Direct 

Projection.      If  wc  again  regard  the  plan  at  the  top  as  the 


PERSPECTIVE   LINES.  77 

plan  of  a  miniature  house,  or  model,  set  six  inches  from 
the  eye  at  S,  and  regard  m  m  as  the  plane  of  the  picture, 
in  contact  with  it,  we  can,  by  drawing  lines  from  every 
point  in  the  plan  to  the  station  point,  find  just  where 
every  point  will  appear  in  the  picture ;  the  horizontal 
dimensions  thus  obtained  can  then  be  transferred  di- 
rectly to  the  picture  in  Fig.  12.  They  are  shown  by 
marks  on  the  lower  side  of  mm,  and  will  be  found 
to  agree  exactly  with  the  dimensions  obtained  from  the 
perspective  plan. 

This  method,  which  is  called  that  of  direct  projection, 
is  often  more  convenient  than  the  other,  especially  when 
the  orthographic  plan  has  previously  been  prepared,  and 
when,  as  in  the  present  case,  the  subject  is  simple. 

But  the  method  of  the  perspective  plan  is  more 
convenient  for  designing  in  perspective,  or  for  Advantages 

of  the  Per- 

makmg  a  perspective  drawing,  as  often  has  spective  pian. 
to  be  done,  from  mere  sketches.  It  takes  up  less  room, 
in  the  vertical  direction ;  it  is  less  laborious,  though  re- 
quiring perhaps  more  knowledge  and  skill ;  and  it  has  the 
advantage  previously  pointed  out, — that  it  enables  the 
position  of  points  at  different  levels  to  be  separately  de- 
termined, by  the  use  of  separate  perspective  plans,  and 
enables  several  successive  drawings  to  be  made,  if  neces- 
sary, without  repeating  the  bulk  of  the  labor,  since  the 
perspective  plan  can  be  made  on  a  separate  piece  of 
paper,  and  used  more  than  once. 

Moreover,  the  points  established  on  a  perspective  plan 
explain  themselves  :  it  is  clear  at  a  glance,  and  after 
any  lapse  of  time,  which  denotes  the  door,  which  the 


78  MODERN   PERSPECTIVE. 

window.  In  working  by  direct  projection  from  the 
orthographic  plan,  on  the  contrary,  it  is  almost  impos- 
sible to  remember  which  point  is  which,  and  much  time 
and  labor  are  lost  by  this  confusion. 

102.  In  making  a  perspective  drawing,  we  generally 
The  modus  ^^  ^^^t  tbc  poiuts  V^  and  V^  as  far  apart  as 
operandi.  ^^  convcnient,  then  strike  a  semicircle  between 
them,  and  take  the  point  S  in  such  a  position  upon  it  that 
S  V^  and  S  Y^  shall  be  parallel  to  the  sides  of  the  object 
to  be  drawn.  The  points  of  distance  and  other  vanishing 
points,  and  the  various  horizons,  can  then  be  determined  ; 
the  perspective  plan  drawn  wherever  is  most  conve- 
nient, and  the  perspective  picture  erected  above  it  by 
the  aid  of  vertical  lines  of  measures.  It  is  to  be  noticed 
tliat,  since  the  vanishing  points  and  traces  depend  solely 
upon  the  direction  of  the  lines  and  planes,  not  upon 
their  position  (11),  the  building  may  be  set  to  the  right 
or  to  the  left  of  the  centre,  below  or  above  the  Horizon, 
according  as  more  or  less  of  the  left-hand  side  or  of  the 
roof  is  to  be  sliown. 

Care  must  be  taken  not  to  confound  the  point  S  with 
the  point  assumed  for  the  front  corner  of  the  perspec- 
tive plan,  which  is  entirely  independent  of  it.  The  re- 
volved lines,  also,  D^  V^,  D^V^  etc.,  (89)  must  not  be 
confounded  with  the  traces  V^  V^  V^  V^  etc. ;  that  is 
to  say,  with  H  L  M,  H  E  JST,  etc. 


CHAPTER  VI. 

THE   POSITION  OF  THE  PICTURE.  —  THE  OBJECT  AT  45°.  — > 
THE   MEASUREMENT   OF   OBLIQUELY   INCLINED   LINES. 

IN  the  last  chapter  we  saw  how,  when  the  real  length 
and  direction  of  the  lines  by  which  an  object  is 
defined  are  known,  the  real  length  and  direction  of  their 
perspectives  may  be  exactly  ascertained.  But  our  in- 
vestigations covered  only  the  cases  of  lines  parallel  to 
the  plane  of  the  picture,  whether  horizontal,  vertical,  or 
inclined,  and  of  horizontal  lines  inclined  to  the  picture. 
It  remains  to  find  out  how  to  apply  a  scale  to  lines  in- 
clined to  the  picture,  that  are  not  horizontal,  such  as 
the  lines  of  the  gables,  and  of  the  hips  and  valleys  of 
the  roofs. 

Plate  V.  shows  how  this  is  done,  and  at  the  same  time 
illustrates  some  other  points  of  interest.     In 

Plate  V. 

this  plate,  as  in  the  previous  plate,  we  have  at 
the  side  a  plan,  on  a  small  scale,  illustrating  the  relative 
position  of  the  spectator  at  the  station  point  S  ;  of  the  ob- 
ject, at  A ;  of  the  plane  of  measures,  at  m  m ;  and  of  the 
plane  of  the  picture,  dX^)}^-  ^^  ^^^^  present  plate  the  posi- 
tion of  the  object  at  A,  and  of  the  spectator  at  S,  is  the 
same  as  in  Plate  IV. ;  but  the  position  of  the  plane  of  the 
picture  and  of  the  plane  of  measures  is  changed,  their 
previous  position  being  indicated  by  dotted  lines.    Their 


80  MUDLKis    PEIISPECTIVE. 

position  is  now  taken  so  that  the  centre  at  V^,  and  the 
vanishing  point  of  45°  at  V-^,  coincide.  That  is  to  say, 
the  principal  lines  of  the  building,  E  and  L,  at  right 
angles  with  each  other,  are  now  at  45°  with  the  plane  of 
the  picture,  and  consequently  with  the  plane  of  meas- 
ures ;  and  the  line  X  that  divides  the  angle,  making 
45°  with  each,  is  accordingly  at  right  angles  with  the 
picture  and  the  plane  of  measures,  and  is  parallel  to  the 
Axis.  V-^,  then,  of  course,  comes  in  the  same  ]3lace  that 
V^  does. 

103.  Now,  in  the  first  place,  this  illustrates  the  point 
that,  the  position  of  the  spectator  and  of  the  object  being 
given,  it  is  purely  a  matter  of  convenience  how  we  take 
Position  of    the  plane  of  the  picture,  and  in  what  direction 

the  plane  of  .       .       ,  -,        -r^, 

the  picture,  its  axis  IS  drawu.  In  Plate  I  \^,  Fig.  11  and 
¥m.  12,  the  axis  was  directed  towards  the  end  of  the  house, 
V^  coming  near  the  corner,  and  the  left-hand  side  of  the 
house  was  less  inclined  to  the  plane  of  the  picture  than  the 
Fig  14.  right-hand  side.  In  Plate  V.,  Fig.  14  and  Fig. 
Fig.  15.  15^  xhQ  axis  is  directed  further  to  the  left,  V^  co- 
inciding in  position  with  V^,  and  the  plane  of  the  picture 
and  plane  of  measures  are  equally  inclined  to  both  sides 
of  the  house,  making^  45°  with  each.  But  of  course  this 
would  not  change  the  appearance  of  the  object  A,  as 
seen  from  S ;  and  if  the  drawings  were  both  made  so  as 
exactly  to  cover  and  coincide  with  it,  the  two  represen- 
tations would  look  exactly  alike  when  viewed  each  from 
its  own  station-point,  so  far  as  concerns  their  main  out- 
lines.    These  are  not  affected  by  the  alterations  and 


THE   POSITION   OF   THE   PICTURE,  ETC.  81 

repairs   the   house   has   undergone   since   we  last   saw 
it. 

104  In  the  second  place,  the  plate  illustrates  the 
point  that  it  is  a  much  simpler  thing  to  make  The  object 

and  the  pic- 

a  drawing  in  perspective  when  the  plane  of  tureat45^. 
the  picture,  as  in  Fig.  15,  is  at  an  angle  of  45°  with 
both  sides  of  the  object  than  when  it  lies  accidentally, 
as  in  Fig.  11. 

For  this  attitude  of  the  object  sets  the  station-point 
at  its  maximum  distance  from  the  picture,  The  symme- 
which  is  obviously  an  advantage.     Tliis  dis-   vauithing 

points  and 

taiice,  S  V^,  is  in  this  case  just  half  the  length  horizons, 
of  the  Horizon  between  V^  and  V^.  Moreover,  since  S 
is  equidistant  from  these  points,  and  V^,  which  is  also 
V-^,  is  half  way  between  them,  it  follows  that  D^  coin- 
cides with  V^,  and  also  that  there  is  another  D-^  coincid- 
ing with  D^ ;  that  D^  and  D^  are  also  equidistant  from 
V^ ;  and,  in  short,  that  the  whole  series  of  vanishing  points 
and  points  of  distance  is  symmetrical  about  H  P  P'  in 
one  direction,  as  it  is  about  the  Horizon  in  the  other. 

105.  From  this  symmetry,  which  is  clearly  exhibited 
on  a  small  scale  in  Fig.  14,  it  follows  that  V^  and  V^ 
are  equally  distant  from  the  Horizon,  that  H  L  M  is 
parallel  to  H  Ft  N'  and  H  R  N'  to  H  L  M',  and  hence  that 
V^  and  V^',  which  lie  at  the  intersection  of  these  traces, 
are  at  an  infinite  distance.  Hence  the  lines  Q  and  Q'  are 
both  parallel  to  the  picture,  the  perspective  of  Q  is  par- 
allel to  H  R  N  and  H  L  M',  that  of  Q'  to  H  L  M  and 
H  R  N',  and  the  projections  of  these  lines  in  the  per- 
spective plan  are  parallel  to  tlie  Horizon. 


82  MODERN   PERSPECTIVE. 

106.  The  great  convenience  of  this  is  shown  in  Fig. 

15,  where  the  hips  of  the  roof  of  the  school- 

The  practical 

conveuieuces  housc  on  the  Icft,  that  form  its  outline  against 

of  this.  ° 

the  sky,  are  drawn  parallel  to  the  horizons  of 
the  planes  in  wliich  they  lie  instead  of  being  directed, 
as  in  Fig.  13,  to  their  inaccessible  point  of  intersection. 

Accordingly  objects  are  generally  drawn  at  45°,  being 
set  to  the  right  or  to  the  left  of  the  centre,  as  it  is  desired 
to  expose  their  left  side  or  their  right  side  more  fully  to 
view  (102). 

107.  It  will  be  observed  that  the  relation  of  the  prin- 
cipal distance  points,  I)^  and  L)^,  to  their  corresponding 
vanishing  points,  V^  and  V^,  is  the  same  as  that  of 
the  corners  of  an  octagon  to  the  corners  of  the  square 
from   which  it  is  cut,  tlie  distance   of   each  from  the 

remoter   corner   beint?   half    the   diagonal   of 

Fig.  16.  °  ° 

the  square.  Compare  Fig.  16  with  the  small 
plan  in  Fig.  14.  The  distance  apart  of  the  vanishing 
points  tlien  being  as  200,  the  distance  of  the  station- 
point  from  the  picture  at  V^  will  be  as  100,  and  each 
point  of  distance  will  be  at  a  distance  of  141  from  its 
own  vanishing  point,  and  59  from  the  other.  Or,  in 
other  words,  the  distance  apart  of  the  vanishing  points 
being  as  10,  that  of  the  points  of  distance  from  the 
centre,  Y*^,  will  be  as  2,  their  distance  from  the  nearest 
vanishing  points  will  be  as  3,  tlieir  distance  from  each 
other  will  be  as  4,  and  the  distance  of  the  spectator 
from  the  picture  as  5,  very  nearly. 

108.  Fig.  17  shows  the  amount  of  the  error  involved 
Fig.  17.        in  this  assumption.     It  is  so  slight  that  for 


nearer  than 
the  plane  of 
measures. 


THE   POSITION   OF   THE   PICTURE,  ETC.  83 

practical  purposes  the  distance  from  the  centre  to  the 
points  of  distance  may  generally  be  taken  as  two  fifths  of 
that  to  the  vanishing  points,  when  the  picture  is  at  angles 
of  45°  with  the  sides  of  the  object. 

109.  Fig.  15  also  illustrates  another  point  of  capital 
importance,  showing  how  dimensions  are  laid  ^^Jj;;J; 
off  by  scale  upon  lines  lying  on  the  hither  side 
of  the  plane  of  measures ;  these  lines  are  necessarily 
drawn  to  a  larger  scale  than  that  used  for  lines  in  that 
plane,  just  as  lines  beyond  it  are  drawn  to  a  smaller  scale. 
The  principle  of  the  isosceles  triangle,  explained  in  the 
last  chapter,  and  again  illustrated  in  Fig.  14,  still  holds 
good.  The  fence,  for  instance,  that  is  built  out  from  the 
front  corner  of  the  house,  toward  the  right,  has  the  posi- 
tion of  its  posts  laid  off  upon  the  line  of  horizontal  mea- 
sures, or  ground  line,  g  I,  to  the  right,  by  scale,  and  then 
transferred  to  the  line  of  the  fence  in  the  perspective 
plane,  by  parallel  lines  drawn  to  D^,  the  fence  being  in 
a  line  with  the  left-hand  side  of  tlie  house.  These  lines 
diverge  as  they  come  forward,  and  the  scale  of  the  fence 
is  obviously  increased.  The  building  on  tlie  left  is  en- 
tirely in  front  of  the  plane  of  measures. 

110.  To  avoid  confusion  the  points  on  the  ground  line 
intended  for  use  beyond  the  plane  of  measures  are  indi- 
cated above  it,  and  those  wliich  give  the  dimensions  of 
objects  this  side  of  it,  as  is  the  case  with  this  fence,  are 
indicated  below  it.  A  better  way  in  some  cases  is  to 
have  separate  ground  lines  for  figures  in  front  of  the 
plane  of  measures. 


84  MODERN   PERSPECTIVE. 

111.  Let  US  now  take  up  the  question  left  unanswered 
Obliquely       jj^  ^j^q  previous  chapter,  and  see  how  dimen- 

inclmed  ^  '- 

Se.^^  sions  can  be  laid  off  by  scale  upon  obliquely 
inclined  lines ;  upon  lines,  that  is  to  say,  which,  like  the 
gables  and  the  hips  and  valleys  of  roofs,  lie  in  inclined 
planes,  and  are  hence  inclined  not  only  to  the  plane 
of  the  picture  but  also  to  the  horizontal  plane. 

In  the  first  place,  it  is  clear  that  just  as  every  hori- 
Line  of  zontal  pLinc  has  a  line  of  horizontal  measures, 
I^^^Yaln     or  ground  line,  wliere  it  intersects  the  plane 

obliquely  ia-  •       t        ^ 

dined  plane,  of  mcasurcs,  and  just  as  every  vertical  plane 
has  a  line  of  vertical  measures  which  is  its  line  of 
intersection  with  the  plane  of  measures,  so  every  in- 
clined plane,  such,  for  example,  as  a  roof,  in  like  man- 
ner has  its  own  line  of  measures  lying  in  it  and  also 
in  the  plane  of  measures,  upon  which  dimensions  can 
be  laid  off  by  scale.  It  is  the  line  in  which  that  plane 
intersects  the  plane  of  measures,  and  it  passes  through 
the  point  at  which  any  line  in  that  plane  pierces  the 
plane  of  measures.  This  line  of  measures  and  its  per- 
spective are  always  parallel  to  the  liorizon  of  the  given 
inclined  plane.  For,  as  we  Lave  seen  (78)  this  horizon  is 
the  intersection  of  the  plane  of  the  picture  with  a  plane 
parallel  to  the  given  plane,  and  passing  through  tlie  eye, 
while  the  line  of  measures  is  the  intersection  of  the 
given  plane,  with  a  plane  parallel  to  the  picture.  The 
intersections  of  these  two  sets  of  parallel  planes  must 
be  parallel. 

112.  To  find  a  line  of  real  measures  in  any  plane, 
then,  it  suffices  to  find  the  point  at  which  any  line  in  it 


THE   POSITION    OF   THE    PICTUKE,    ETC.  85 

pierces  the  plane  of  measures,  and  to  draw  through  this 
point,  when  found,  a  line  parallel  to  the  horizon  of  the 
plane. 

113.  In  Fig.  15,  for  example,  we  have  several  such 
lines  :  g  I  parallel  to  the  Horizon,  I  m  parallel 

to  H  L  M,  r  n  parallel  to  H  E  N,  etc.  In  the 
case  of  the  ground  line,  the  horizontal  lines  to  be  meas- 
ured off  are  prolonged  until  they  reach  the  plane  of  meas- 
ures, and  the  line  g  I  is  drawn  parallel  to  the  horizon 
through  the  point  thus  obtained.  In  the  case  of  the  in- 
clined lines  of  the  gables,  that  on  the  end  of  the  building 
at  the  right  is  brought  forward  till  it  touches  the  plane 
of  measures,  and  the  corresponding  line  on  tlie  porch  roof 
of  the  porch  of  the  little  school-house  on  the  left  is  carried 
back  until  it  reaches  the  plane  of  measures.  These  points 
are  ascertained  from  the  perspective  plan,  tlie  point  where 
the  horizontal  projection  of  these  inclined  lines  pierces 
the  plane  of  measures  being  the  projection  of  the  point 
where  the  lines  themselves  do  so.  The  lines  of  meas- 
ures I  m,  are  tlien  drawn  through  the  points  thus  ascer- 
tained parallel  to  tlie  horizon  of  the  plane  L  M.  As 
these  gable  lines  lie  also  in  the  vertical  planes  E  Z, 
vertical  lines  of  measures,  r  z,  parallel  to  H  E  Z,  may 
also  be  drawn  through  the  same  points. 

114.  In  the  case  of  the  hip  line,  Q,  of  the  little 
school-house,  which  is  parallel  to  the  plane  of  measures 
and  to  the  horizon  HEN  (105),  it  is  necessary  to  pro- 
ceed as  in  the  case  of  vertical  lines,  or  any  others  that  are 
parallel  to  the  picture,  and  carry  across  it  a  line  lying 
also  in  the  pkne  E  N,  but  inclined  to  the  picture,  by 
means  of  which  the  line  where  the  plane  intersects  the 


86  MODERN   PERSPECTIVE. 

plane  of  measures  can  be  found.  In  the  figure  the  line 
of  the  other  hip,  P,  is  used  for  this  purpose,  its  horizon- 
tal projection  being  found  in  the  perspective  plan.  It 
would  have  done  just  as  well  to  prolong  the  line  of  the 
eaves  until  it  met  the  plane  of  measures,  and  to  draw  the 
line  rn  through  the  point  thus  found. 

115.  All  these  lines  of  measures  lie  in  the  plane  of 
measures,  and  are  consequently  drawn  to  the  same  scale. 
On  all  of  them  one  sixteenth  of  an  inch  is  the  true 
perspective  representation  of  one  foot,  and  any  dimen- 
sions laid  off  on  this  scale  can  be  transferred  to  any  other 
lines  in  the  inclined  plane,  such  as  the  lines  of  the 
gables,  by  means  of  proper  points  of  distance  lying  in 
the  horizon  of  the  inclined  plane,  just  as  dimensions  can 
be  transferred  from  the  ground  line  to  lines  lying  in  the 
horizontal  plane  by  means  of  points  of  distance  taken 
on  the  Horizon. 

116.  The  different  points  of  distance  upon  these 
Manifold       inclined    horizons,    are   easily  obtained    from 

points  of  dis- 
tance, those  upon  the  horizontal  Horizon,  if  we  may 

use  such  an  expression.  It  is  obvious,  indeed,  that  the 
problem  is  a  simple  one,  if  we  remember  that  the  point 
of  distance  corresponding  to  any  vanishing  point  is 
found  by  setting  off  upon  the  horizon  passing  through 
that  point  the  true  distance  of  that  point  from  the  sta- 
tion-point. If  this  is  done,  the  triangle  S  V  D  formed 
by  the  three  points  will,  of  course,  always  be  isosceles 
by  construction.  Now,  the  distance  of  every  vanishing 
point  from  the  eye  is  known,  and  can  readily  be  laid  off 
on  all  the  traces,  or  horizons,  that  pass  through  it. 


THE   POSITION   OF   THE   PICTURE,    ETC.  87 

117.  The  distance  of  V^  from  S,  for  example,  is 
known  to  equal  V^  D^,  and  this  distance  laid  The  io(nisot 

points  of  dis- 

off  on  H  R  N,  H  E  N',  H  R  Z,  and  the  traces  tance. 
of  any  other  planes  that  contain  R,  give  the  corresponding 
points  of  distance  on  those  horizons,  D^\  D^'^,  D^,  D^, 
etc.,  all,  in  fact,  lying  at  equal  distances  about  V^  on 
the  circumference  of  a  circle  whose  radius  equals  V^  S, 
the  distance  of  the  vanishing  point  in  question  from  the 
eye.     This  is  shown  both  in  Fig.  14  and  in  Fig.  14. 
rig.    15.     In   like    manner   a   circle  may  be  ^ig.  15. 
drawn   about   V^,  containing  the  distance-points   that 
belong  to   that   vanishing  point.     Its   radius   will   be 
V^  D^  the  distance  of  V^  from  the  eye. 

Such  a  circle  is  the  locus  of  the  points  of  distance. 

118.  The  distance  of  \^  from  the  eye  is  Y^^  D^,  as  is 
obvious  if  we  suppose  the  triangle  V^  D^  V^  revolved 
back  around  its  side  H  R  Z  until  D^  is  at  S  again.  A 
circle  drawn  around  V^,  with  V^^  D®=Y^  S  as  a  radius, 
will  accordingly  give  points  of  distance,  by  which  any 
inclined  line  M,  vanishing  at  Y^^,  may  be  measured  off 
by  lines  of  measures  lying  in  any  of  the  planes  that  con- 
tain it,  and  whose  horizons  accordingly  pass  through  Y^. 

As  we  have  taken  the  lines  M  and  M'  at  an  angle  of 
30°  with  the  Horizon,  the  angle  M  D«  M'  is  60°,  the 
triangle  M  D^  M^  is  equilateral,  and  D^  falls  upon  Y^', 
and  D^'  upon  Y^. 

In  every  case,  of  course,  the  line  of  real  measures, 
like  the  lines  of  proportional  measures  discussed  in  a 
previous  chapter,  is  drawn  parallel  to  the  horizon  of  the 
plane  containing  it  (76). 


88  MODERN   PERSPECTIVE. 

119.  It  will  be  noticed  that  there  are  two  points  of 
Two  points  of  distance  on  H  E  Z,  at  equal  distances  above 
each  horizon,  and  bclow  Y^,  just  as  wc  foiuid  that  both  V^ 
and  V^  would  serve  as  points  of  distance  to  V-^.  Either 
of  these  can  be  used  instead  of  D^^  to  set  of!'  given 
lenoths  on  the  rioht-hand  horizontal  lines.  In  the 
upper  perspective  plan,  for  example,  the  length  of  the 
right-hand  side  of  the  house  is  set  off  not  only  on  the 
ground-line,  and  transferred  by  means  of  D^,  but  also 
on  the  vertical  line  of  measures,  both  above  and  below 
the  ground,  and  transferred  by  means  of  D^^  and  D^, 
giving  in  every  case  the  same  points. 

Indeed,  it  is  practicable  to  have  a  second  point  of 
distance  on  all  the  horizons  beyond  V^.     They 

Fig.  18.  *^ 

would  be  the  vanishing  points  of  the  bases  of 
ohtuse-angled  isosceles  triangles.  Such  a  triangle  is  shown 
in  Fig.  18,  h.  Its  employment  is  shown  in  the  upper  per- 
spective plan,  where  the  length  of  the  right-hand  side 
of  the  house  is  laid  off  by  scale  on  g  /,  to  the  left  of  the 
corner,  and  transferred  to  the  perspective  line  by  means 
of  the  point  D^*,  which  is  out  of  the  picture. 

120.  If,  then,  through  the  point  at  which  any  per- 
spective line  pierces  the  plane  of  measures,  a  line  of 
measures  be  drawn  parallel  to  the  horizon  of  any  plane 
that  contains  the  line  (that  is  to  say,  parallel  to  any 
horizon  passing  through  the  vanishing  point  of  the  line), 
any  distance  laid  off  by  scale  on  the  line  of  measures 
may  be  transferred  to  the  perspective  line  by  means  of  a 
point  of  distance  in  that  horizon  ;  and  this  point  of  dis- 
tance will  be  the  point  where  that  horizon  is  cut  by  the 


THE   POSITION    OF   THE   PICTURE,  ETC.  89 

circle,  which  is  the  locus  of  the  points  of  distance,  —  a 
circle  whose  radius  equals  the  distance  of  the  vanishing 
point  of  the  perspective  line  from  the  eye,  and  whose 
centre  is  at  the  vanishing  point. 

This  is  illustrated  in  the  lower  perspective  plan,  in 
which  a  certain  distance,  being  that  from  the  corner  of 
the  house  to  the  further  edge  of  the  window,  is  laid  off 
on  several  lines  of  measures,  gl,  Iz,  and  I  m',  parallel  to 
the  several  traces  that  meet  at  Y^.  Lines  drawn  to  the 
points  D^  on  those  traces  all  give  the  same  point  upon 
the  line  L. 

121.  As  the  points  taken  upon  the  line  of  measures 
are  all  equidistant  from  the  corner  of  the  ,^^^  ^^^^^^  ^^ 
plan,  they  lie  in  the  circumference  of  a  small  ei'lmf  mLs- 
circle,  which  is  the  locus  of  the  points  of  meas- 
ures, just  as  the  corresponding  points  of  distance  lie  in 
the  circumference  of  a  large  circle ;  and  we  have  the 
curious  phenomenon  of  lines  drawn  from  certain  points 
on  tlie  small  circle  to  corresponding  points  on  the  large 
one,  all  intersecting  at  the  same  point,  upon  a  line  join- 
ing the  two  centres.  But  Fig.  19,  which  pre-  rig.  19. 
sents  these  geometrical  relations  in  a  diagram,  shows 
that  there  is  nothing  surprising  in  this. 

122.  Now,  just  as  any  line  drawn  in  any  direction,  at 
random,  through  one  end  of  a  perspective  line,   ^^^^0^ 
may  be  used  as  a  line  of  proportional  meas-  equal ""^ 
ures  (81),  so  any  such  line   that  lies  in  the 

plane  of  measures  may  be  taken  as  a  line  of  real  meas- 
ures ;  and  any  dimensions  taken  upon  it  by  scale  may 
be  transferred  to  any  perspective  line  tliat  touches  it, 


90  MODERN    PERSPECTIVE. 

by  means  of  a  horizon  drawn  through  the  vanishing 
point  of  the  perspective  line,  parallel  to  the  random  line, 
the  point  of  distance  being  taken  upon  that  horizon  at 
the  same  distance  from  that  vanishing  point  as  its  otlier 
points  of  distance. 

This  also  is  illustrated  in  the  lower  persj)ective  plan, 
where,  in  addition  to  the  lines  of  measures  drawn  par- 
allel to  the  horizons  already  obtained,  a  line  of  measures, 
It,  is  drawn  at  random  at  an  angle  of  15°.  A  new  hori- 
zon, HLT,  is  drawn  through  V^  parallel  to  it,  and  a 
new  point  of  distance,  D^^,  obtained  upon  it,  where  it 
intersects  the  locus  of  D^ 

The  same  thing  is  done  again  in  the  picture  Fig.  15, 
Fig.  15.  where  the  divisions  of  the  fence,  on  the  right 
of  the  house,  are  taken  by  scale  on  an  arbitrary  line  of 
measures  just  above.  As  this  also  is  taken  at  15°,  the 
same  point  of  distance,  D^^,  serves  to  transfer  the  dimen- 
sions to  the  line  L,  the  top  line  of  the  fence. 

123.  Finally,  just  as  of  any  two  perspective  lines 
Pers  ective  ^^^^^"0  ^^^  samc  Vanishing  point,  one  may  be 
uCTry'ho^r'  regarded  as  the  horizon  of  a  plane  passing 
through  the  other  (82),  so  a  point  of  distance 
taken  upon  the  first,  at  the  proper  distance  from  that 
vanishing  point,  may  be  used  to  lay  off  given  dimensions 
upon  the  second,  by  using  a  scale  upon  a  line  of  real 
measures  drawn  parallel  to  the  first,  from  the  point 
where  the  second  pierces  the  plane  of  measures. 

This  is  illustrated  in  the  plan  of  the  little  school- 
house  shown  in  the  plate.    The  front  line  of  the  plan  is 


I 


THE   POSITION    OF   THE    PICTURE,   ETC.  91 

regarded  as  the  horizon  of  a  plane,  and  upon  this  horizon 
the  point  of  distance  D^,  upon  the  locus  of  D^,  is  easily- 
determined.  A  line  of  measures  is  drawn  parallel  to  it 
through  the  further  corner  of  the  plan,  where  the  line  E 
on  the  back  of  the  building  touches  the  plane  of  meas- 
ures. The  true  width  of  the  building  being  laid  off  on 
this  line  of  measures  by  scale,  and  transferred  to  the 
perspective  of  the  further  side  by  means  of  a  line  drawn 
to  the  point  of  distance,  we  obtain  the  left-hand  corner  of 
the  building,  and  the  position  of  the  rear  window,  as  we 
ouo-ht  to. 

o 

124.  Here,  as  before  (84),  we  must  be  careful  not  to 
suppose  that  this  line  of  measures  is  drawm  on  the  floor  of 
the  school-house.  It  lies,  in  fact,  in  the  vertical  plane 
of  measures,  being  an  inclined  line  parallel  to  the  plane 
of  the  picture.  So,  also,  the  point  of  distance  is  not  on 
the  ground,  but  is  in  the  infinitely  distant  horizon  which 
the  front  line  of  the  building  covers  and  apparently  co- 
incides with. 

125.  It  is  to  be  observed  that  the  methods  of  mea- 
suring off  perspective  lines  described  in  this 

Proportional 

and  the  preceding  chapters  are  strictly  analo-  ^^''aTme^ 
gous  to  the  method  of  dividing  perspective   srak'QTea- 
lines  described  in  the  previous  chapter.     By 
that  method,  called  the  Metliod  of  Triangles,  the  per- 
spective of  a  finite  line  is  divided  up,  in  any  desired  pro- 
portion, by  drawing  from  one  end  of  it  a  line,  called  a 
line  of  proportional  measures,  parallel  to  the  plane  of 
the  picture,  and  then  setting  off  upon  this  Line,  at  any 


92  MODERN   PERSPECTIVE. 

convenient  scale,  dimensions  proportional  to  those  de- 
sired. These  are  then  transferred  to  the  perspective  of 
the  given  line  by  means  of  a  point  of  proportional  mea- 
sures which  is  taken  upon  the  horizon  of  the  plane  in 
which  both  lines  lie.  This  horizon  passes  through  the 
vanishing  point  of  the  given  line,  and  the  point  of  pro- 
portional measures  is  the  vanishing  point  of  the  base  of 
the  scalene  triangle,  the  other  two  sides  of  which  are 
the  given  line  and  the  line  of  measures. 

By  using  the  point  of  distance,  however,  as  a  point  of 
measures,  the  triangle  becomes  isosceles,  and  the  parts  cut 
off  upon  the  perspective  line  are  not  only  proportional 
but  equal  to  those  taken  upon  the  line  of  measures.  It 
is  now  the  line  of  measures  that  is  of  fixed  dimensions, 
and  the  perspective  line  is  not  divided  up,  but  has  these 
dimensions  set  off  upon  it,  being  treated  as  a  line  of 
indefinite  length.  In  the  Method  of  Triangles  the  line 
to  be  divided  is  of  definite  length,  the  line  of  propor- 
tional measures  is  an  indefinite  line,  and  the  point 
of  (proportional)  measures  may  fall  anywhere  in  the 
horizon  of  the  plane  of  the  triangle.  In  measuring  a 
given  distance  upon  a  perspective  line,  on  the  other 
hand,  the  line  of  equal  measures,  or  of  scale  measures 
if  a  scale  is  used,  is  a  definite  portion  of  the  ground  line, 
or  other  line  of  measures,  the  given  line  is  of  indefinite 
length,  and  the  point  of  (equal)  measures  is  the  point 
of  distance. 

In  dividing  a  given  line  proportionally,  the  length  of 
the  line  is  indeed  always  fixed,  and  tliat  of  the  auxiliary  is 
indefinite  ;  in  transferring  real  dimensions,  on  the  other 


THE   POSITION   OF   THE   PICTURE,   ETC.  93 

hand,  these  fix  the  length  of  the  auxiliary,  and  it  is 
upon  an  indefinite  length  of  given  line  that  they  are 
to  be  set  off.  This  difference  is  illustrated  in  Fig.  18, 
Plate  V.  In  a  the  given  line  is  of  definite  length, 
the  auxiliary  of  indefinite  ;  in  h  the  contrary  is  the 
case. 

Moreover,  it  is  to  be  observed  that  although  a  line  of 
equal  measures,  like  a  line  of  proportional  measures,  may 
be  drawn  parallel  to  the  picture  from  any  point  in  the 
perspective  line,  it  will  not  serve  for  scale  measure- 
ment unless  it  lies  in  a  plane  of  measures  whose  posi- 
tion has  been  already  determined.  It  is  accordingly 
necessary  to  extend  the  given  line  until  it  touches  or 
pierces  the  established  plane  of  measures,  before  a  line 
of  scale  measures  can  be  drawn,  as  has  frequently  been 
exemplified  in  this  chapter.  The  difference  is  illus- 
trated by  comparing  the  treatment  of  the  gable  in 
Plate  III.  with  tliat  of  the  gable  in  Plate  V.  In  the 
latter  the  line  of  the  gable  is  extended  so  as  to  bring 
the  lines  of  measures  into  the  same  plane  as  all  the 
other  lines  of  measures. 


chapte:r  yii. 

PARALLEL    PEESPECTIVE.  —  CHANGE   OF   SCALE. 

THE  last  chapter  discussed  the  case  in  which  the 
plane  of  the  picture,  and  consequently  the  plane 
of  measures  parallel  to  it,  is  set  at  an  angle  of  ^^5°  with 
the  sides  of  the  object,  so  that  the  "vanishing  point  of 
45°,"  V-'^,  coincides  with  V^^  the  centre  of  the  picture, 
and  the  principal  vanishing  points  of  the  right-hand  and 
left-hand  lines,  V^  and  V^,  and  their  points  of  distance, 
D^  and  D^,  are  symmetrically  disposed  on  each  side  of  it. 
The  diagonal  Y,  at  right  angles  to  X,  is  in  this  case, 
of  course,  parallel  to  the  picture,  and  to  the  Horizon. 

126.  Let  us  now  consider  the  analoij^ous  case  in  which 

the  plane  of  tlie  picture,  and  consequently  the 
plane  of  measures,  is  taken  parallel  to  one  of  the 
principal  sets  of  lines,  and  at  right  angles  to  the  other. 
This  is  illustrated  in  Plate  VI.  When  objects  are  thus 
represented  with  one  side  parallel  to  tlie  picture,  and 
the  adjacent  sides  perpendicular  to  it,  they  are  said  to 
be  drawn  in  Parallel  Perspective. 

127.  The  relation   between  this  case  and  that  dis- 

cussed in  the  last  paper  is  shown  in  the  two 

Fig.  20. 

buildings  upon  the  quay,  on  the  left-hand  side 


PARALLEL    PERSPECTIVE.  95 

of  Fig.  20.  The  nearest  one,  whose  roof  rises  just  above 
the  rail  of  the  descending  steps,  stands  at  45°  with  the 
picture,  just  as  in  Plate  V.,  having  the  vanishing  points 
of  its  main  lines  at  V^  and  V^;  while  one  set  of  its  diag- 
onals, X,  as  seen  in  the  perspective  plan  below,  converges 
at  Y^,  and  the  other,  Y,  is  parallel  to  the  Horizon  ;  the 
lines  of  the  hips,  P  and  P',  being  directed  to  V^  and  V^', 
and  the  hips  Q  and  Q'  being  parallel  to  the  horizons  H  RN 
and  H  L  M  (105),  The  points  V^^  and  V^  are  off  the  pic- 
ture, but  Y^  suffices  to  determine  the  position  of  these 
horizons.  These  horizons  are  not  shown  in  the  picture,  to 
avoid  confusion.    The  diagonal  X  here  coincides  with  C. 

128.  The  larger  building  beyond,  like  all  the  other 
objects  in  the  picture,  is  drawn  in  Parallel  paraiieiper- 
Perspective.  It  is  set  at  an  angle  of  45°  "P"""""- 
with  the  building  just  mentioned,  the  sides  of  one  being 
parallel  to  the  diagonal  lines  that  divide  the  angles  of 
the  otlier,  and  vice  versa.  This  is  rendered  more  obvious 
by  comparing  their  perspective  plans. 

129.  These  plans,  it  will  be  noticed,  are  drawn  wher- 
ever it  is  most  convenient  to  put  them,  that  of  the  fur- 
ther building  being  taken  so  very  far  below  the  ground 
as  to  come  lower  down  on  the  paper  than  that  of  the 
nearer  building.    The  diagonal  X  here  coincides  with  R. 

130.  Now,  it  is  to  be  observed  that  while  the  planes 
of  the  nearer  roofs,  as  in  the  previous  plates, 

have  for  their  horizons  the  lines  H  R  N,  H  L  M,  SjfJ'eJm'Int' 
etc.,  and  the  hips  have  their  vanishing  points  Sormauo'^ 
at  Y^,  Y^',  etc.,  the  roof  of  the  further  build-  '^^p'^*"'-^- 
ing,  and  the  tops  of  the  posts  in  the  foreground,  which, 


96  MODERN   PERSPECTIVE. 

for  convenience,  are  given  the  same  slope,  present  a  new 
case,  which  we  have  not  hitherto  met,  —  the  case  of 
inclined  planes  whose  horizontal  element  is  either  par- 
allel or  perpendicular  to  the  picture. 

131.  The  several  flights  of  steps,  also,  ascend  and 
descend  along  inclined  planes,  either  at  right  angles  to 
the  picture,  or,  as  is  the  case  of  those  at  either  end  of 
the  platform  in  the  extreme  foreground,  parallel  with  it. 
Let  us  take  these  flights  of  steps  first,  and  make  the  rise 
of  each  step  six  inches,  and  the  tread  nineteen  inches. 
Now,  as  the  flights  in  the  foreground  are  parallel  to  tlie 
picture,  they  will  be  drawn  in  their  true  proportions, 
and  will  give  the  true  slope  of  the  steps ;  and  if  we 
suppose  the  plane  of  measures  to  coincide  with  the  front 
of  the  little  pavilion,  or  with  the  further  end  of  the 
stej)s,  we  can  lay  off  the  steps  by  scale  at  once,  and 
ascertain  their  true  slope.     It  proves  to  be  about  20°. 

132.  To  ascertain  the  point  V-^,  which  determines  the 
direction  of  the  slopiug  lines  of  the  flights  which  ascend 
at  right  angles  to  the  picture,  we  have  only  to  draw  a 
line  at  V^,  =  D^,  making  an  angle  of  20°  with  the  Hori- 
zon. Its  point  of  intersection  with  H  P  P',  above  V^, 
will  be  the  point  in  question  (88).  V^',  the  vanishing 
point  of  the  descending  flights,  will  be  at  an  equal 
distance  below  V^. 

133.  The  horizons  of  the  inclined  planes  in  which 
these  steps  lie,  pass,  of  course,  through  the  vauisliing 
points  of  all  the  lines  that  lie  in  them  (13,  II.).  V^  and 
V^'  are  the  vanishing  points  of  the  steepest  lines  of 
the  ascending  and  descending   planes,  their  horizontal 


PARALLEL   PEKSPECTIVE.  97 

element  K  being  parallel  to  the  picture,  and  tlieir  vanish- 
ing points  being  at  an  infinite  distance  to  the  right  and 
left.  The  horizons  drawn  through  Y^  and  V^'  are 
accordingly  drawn  horizontal,  or  parallel  to  the  Horizon. 
The  horizon  of  a  plane  is,  indeed,  always  parallel  to  that 
element  of  the  plane  whicli  is  parallel  to  the  plane  of 
the  picture  (76);  for  that  element  has  its  vanishing  point 
on  the  horizon,  but  at  an  infinite  distance.  These  two 
horizons  are  H  K  A  and  H  K  A',  just  as  the  Horizon  is 
lettered  H  R  L. 

134.  The  horizon  of  the  inclined  planes  of  the 
flights  of  steps  that  ascend  to  the  right  and  go  down 
to  the  left,  on  the  edges  of  the  pictui'e,  passes  of 
course  through  the  point  V^,  the  vanishing  point  of 
their  horizontal  element,  and  is  parallel  to  the  steepest 
line  of  the  slope,  S,  which  is  the  element  parallel  to  tlie 
picture. 

135.  The  horizons  of  the  inclined  planes  of  the  roof  of 
the  building  on  the  left,  and  of  the  little  flat  pyramids 
on  top  of  the  posts,  are  determined  in  the  same  way. 
The  slope  of  these  planes  is  about  25° ;  they  are  accord- 
ingly steeper  than  the  slope  of  the  steps ;  and  the  van- 
ishing points  V^2  ^i^d  ^Vj  found  by  drawing  lines  from 
yL  Qj^.  yR  j^|-  r^^-^  angle  of  25°,  are  further  from  the  Horizon 
than  V^  or  V^'.  H  K  A2  and  H  K  A.2'  are  the  horizons  of 
the  front  and  back  planes,  while  the  planes  of  the  right- 
hand  and  left-hand  sides  pass  diagonally  across  the  pic- 
ture through  Y^  at  an  angle  of  25°.  They  are  lettered 
H  C  D  and  H  C  SI  The  vanishing  points  of  the  hips,  or 
angles  of  the  pyramids,  are  at  the  intersections  of  these 

7 


98  MODERN    PERSPECTIVE. 

horizons  with  H  K  A2  and  H  K  A2'  at  the  points  marked 
YM2^  YN2^  YM2/^  ymt  They  are,  of  course,  in  the  hori- 
zons H  LZ  and  H  E  Z,  since  these  hips  and  angles  obvi- 
ously lie  in  vertical  planes  making  45°  with  the  picture. 

136.    The  only  other  object  in  the  figure,  the  pavilion 
or  belvedere,  a  little  building  just  twice  as  long 

Dimensions.  .   ,   .  .      ,  .  ,        .  •it  a  n    j_i 

Within  as  it  is  wide,  is  easily  drawn.  All  the 
lines  of  the  perspective  plan  are  either  parallel  to  the 
Horizon  or  converge  to  V^,  and  are  cut  off  at  the  length 
required  by  setting  off  the  true  length  by  scale,  upon 
ff  I,  and  transferring  it  to  the  perspective  line  by  a 
line  drawn  to  the  point  of  distance,  V^,  =  D*^.  The 
intermediate  points  are  determined  in  the  same  way.  The 
front  of  the  building  being  in  the  plane  of  measures,  all 
its  parts  are  drawn  to  scale,  proportionally  to  their  real 
dimensions,  and  the  same  is  true  of  the  further  end  of 
the  building  and  of  the  arched  wall  in  the  middle,  only 
that  being  more  distant,  the  scale  on  which  they  are 
drawn  is  smaller.  All  the  arches  are  struck  from  the  per- 
spective of  their  centres,  which,  being  in  reality  all  on  a 
horizontal  line  perpendicular  to  the  picture,  occur  in  the 
perspective  on  a  line  drawn  from  the  centre  of  the  front 
arch  to  V^.  Their  exact  position  on  that  line  is  deter- 
mined on  the  perspective  plan. 

In  the  same  way  the  rafters  in  the  roof  may  be  laid 
off  exactly,  two  feet  apart,  using  the  point  of  distance 
D^  =  V^,  or,  if  their  number  is  known,  the  space 
they  occupy  may  be  divided  into  six  equal  parts  by  using 
the  method  of  triangles,  described  in  the  fourth  chapter. 


PARALLEL   PERSPECTIVE.  99 


137.  Fur  the  length  indicated  in  the  figure,  and  with 
the  station  point  so  near  the  picture,  there  is  cjiangeof 
no  practical  inconvenience  in  thus  using  V^  or  ^*^^^^* 

V^  as  points  of  distance.  But  if,  as  niiglit  easily  be,  the 
room  to  be  drawn  were  twice  as  long  as  this,  the  point 
g  would  be  inconveniently  far  away ;  and  if  the  station 
point  were  farther  from  the  picture,  —  and  it  is  always  an 
object  to  have  the  station  point  as  far  away  as  the  point 
from  which  the  picture  will  generally  be  regarded, — 
say  two  feet,  the  point  Y^  also  would  be  practically  in- 
accessible. 

138.  These  inconveniences,  which  are  likely  to  occur 
in  oblique  perspective  as  well  as  in  parallel  perspective, 
may  in  all  cases  be  got  over  by  substituting  another  tri- 
angle for  the  isosceles  triangle  hitherto  employed.  In- 
stead of  using  a  triangle  whose  legs  are  equal  we  may 
just  as  well  employ  a  scalene  triangle,  provided  only  the 
ratio  of  the  two  legs  is  known.  Lines  drawn  parallel  to 
the  base  will  not  now  indeed  divide  the  adjacent  sides 
into  equal  parts,  but  we  can  just  as  easily  as  before  cut 
off  any  required  dimensions. 

139.  This  is  illustrated  in  Fig  21,  c.    We  have  here,  as 
in  Fig.  12  and  Fig.  14,  the  plane  of  the  picture     Fig.  21. 
pphi  immediate  contact  with  the  object,  which     ^j^  12. 

is  here  the  model  of  a  small  room  or  passage,  ^'^' 
whose  plan,  as  above,  occupies  two  squares.  Let  us 
suppose  the  spectator  to  be  at  Si,  at  a  distance  from 
the  picture  of  Si  D^,  the  length  of  the  Axis,  C^.  V^ 
then,  or  D^  will  be  at  an  equal  distance  to  the  right, 
as  shown,  on  the  prolongation  of  ][)  p.     This  gives  the 


100  MODERN   PERSPECTIVE. 

right-angled  isosceles  triaDgle,  Sj  V^  D^,  and  the  length 
of  the  room  laid  off  on  ^  ^  in  the  other  direction  may 
be  transferred  to  the  side  of  the  room  by  the  line  D  d, 
drawn  parallel  to  Si  I)^.     Fig.  21,  h,  shows  this  done  in 

perspective,  determining  the  point  d^  as  just 

now  in  Fig.  20. 

140.  But  this  point  d  will  be  fixed  with  equal  pre- 
cision if  we  take  instead  of  D^  another  point,  Di,  half 
way  from  V^  to  D^  and  make  the  base  of  our  triangle 
Si  Di.  The  triangle  is  no  longer  isosceles,  but  we  know 
that  lines  drawn  parallel  to  the  base  will  make  the  seg- 
ments of  the  short  side  just  half  as  long  as  those  of  the 
long  side,  and  that  does  just  as  well.  For  if  we  now 
lay  off  in  the  other  direction,  just  as  before,  half  the 
length  desired,  a  line  drawn  parallel  to  Si  Di  will  give 
fZ,  just  as  before.  And  in  like  manner  we  might  take 
Di,  half  way  between  V^  and  Di,  and  lay  off  one  quar- 
ter the  required  length  of  the  room,  etc.,  with  the  same 
result. 

141.  Applying  this  now  to  Fig.  20,  by  taking  Di  or 

Di,  measured  off  alonf?  the  Horizon  from  V^  at 

Fig.  20  ""  . 

half  or  quarter  the  distance  of  the  station  point 
from  that  vanishing  point,  we  can  cut  off  any  desired  di- 
mensions on  the  perspective  lines  that  converge  at  V^  by 
laying  them  off  upon  the  ground  line,  ^  /,  by  a  scale  of 
equal  parts  half  or  quarter  as  large  as  those  used  in  the 
plane  of  measures,  and  employed  for  the  horizontal  and 
vertical  lines  in  that  plane.  Instead  of  using  a  half- 
inch  scale,  and  using  D^  as  the  point  of  distance,  we 
may  use  a  quarter-inch  scale,  and  transfer  the  dimen- 


PARALLEL   PERSPECTIVE.  101 

sions  to  C  by  means  of  Di,  or  an  eighth  scale  and 
use  Dx. 

142.  This  results  may  be  summed  up  as  follows :  — 
Auxiliary  points  of  distance,  which  may  be   called 

points  of  half  distance,  quarter  distance,  etc.,  Points  of 

^  .  ^  half  dis- 

may be  obtained  by  laying  off  from  the  van-  tance,  etc. 

ishing  point  of  any  line,  upon  any  horizon  that  passes 
through  that  vanishing  point,  a  half,  or  a  quarter,  etc., 
of  the  real  distance  of  the  station  point  from  that  van- 
ishing point;  the  required  dimension  must  then  be 
laid  off  in  an  opposite  direction  upon  a  line  of  half  or 
quarter  measures,  etc.,  drawn  parallel  to  that  horizon 
through  the  point  where  the  perspective  line  in  ques- 
tion touches  the  plane  of  measures,  the  scale  employed 
being  a  scale  of  equal  parts,  half,  or  quarter,  etc.,  as  large 
as  those  employed  for  lines  in  the  plane  of  measures. 

143.  It  is  sometimes  convenient  also  to  use  a  smaller 
scale  for  lines  parallel  to  the  plane  of  measures  and  at 
some  distance  behind  it,  whether  vertical,  horizontal,  or  in- 
clined. This  is  tantamount  to  establishing  another  plane 
of  measures,  and  another  ground  line,  two,  three,  or  four 
times  as  far  off,  and  diminishing  the  scale  used  in  the 
picture  accordingly. 

144.  This  use  of  points  of  half  distance,  third  dis- 
tance, or  quarter  distance,  and  this  employment  of  an 
auxiliary  plane  of  measures,  and  the  change  of  scale  in- 
volved in  both  these  devices,  are  obviously  just  as  prac- 
ticable in  other  cases  as  they  are  in  this.  But  they  are 
most  often  used  in  the  case  of  parallel  perspective.  For 
in  oblique  perspective  the  need  of  having  the  vanishing 


102  MODERN   PERSPECTIVE. 

points  within  convenient  distance  generally  limits  the 
distance  of  the  station  point  and  keeps  the  points  of  dis- 
tance near  at  hand.  Both  are  always  nearer  than  the 
remoter  vanishing  point.  But  in  parallel  perspective  one 
of  the  principal  vanishing  points  is  at  an  infinite  dis- 
tance, and  the  points  of  distance,  though  nearer  than 
that,  may  yet  be  quite  out  of  reach.  The  use  of  points 
of  half  and  quarter  distance,  etc.,  enables  one  to  set  the 
station  point  as  far  away  as  he  pleases.  There  is 
absolutely  nothing  to  prevent  his  taking  the  point  of 
view  most  favorable  for  his  purpose,  the  point  of 
view,  namely,  that  will  give  the  best  proportions  to  his 
picture. 

145.    In  practice  it  is  most  convenient  to  assume  the 
An  inverse     desivcd  pwportions  at  the  outset ;  that   is   to 

procedure 

common.  say,  having  determined  on  the  scale  at  which 
the  nearest  end  of  the  street  or  room  is  to  be  drawn, 
to  make  the  further  end  as  large,  and  to  set  it  as  far  on 
one  side  and  as  far  up  or  down,  as  will  look  best.  Van- 
ishing lines  drawn  through  the  corresponding  points  of 
the  two  ends  will  then  determine  the  centre,  V^,  and  the 
Horizon.  This  is  all  that  need  be  determined,  since  the 
length  of  the  room  or  street  is  supposed  to  be  known. 
If  from  the  near  end  of  the  perspective  of  this  length, 
the  real  length  is  laid  off  upon  the  ground  line,  at  any 
convenient  scale,  and  the  last  point  connected  with  the 
further  end  of  the  perspective  line,  and  prolonged  until 
it  meets  the  Horizon,  the  point  thus  ascertained  will  be 
a  point  of  half,  quarter,  or  third  distance,  according  as 


PARALLEL   PERSPECTIVE.  103 

the  scale  chosen  is  a  half,  a  quarter,  or  a  third  of  that 
used  for  the  plane  of  measures.  The  corresponding  dis- 
tance of  the  station  point  in  front  of  the  centre,  V^,  will 
then  be  two,  three,  or  four  times  the  distance  of  the  cen- 
tre from  this  auxiliary  distance  point. 

The  two  views  of  the  street  in  Fig.  22,  both  of  which 
were  drawn  in  this  way,  illustrate  the  impor-  Fig.  22. 
tance  of  carefully  proportioning  the  parts  of  the  picture. 
The  upper  one  shows  the  street  as  it  would  appear  quite 
near  at  hand,  much  reduced ;  the  other,  drawn  to  the 
same  scale,  shows  how  it  would  look  at  a  greater  dis- 
tance. 

146.  Inasmuch  as  the  relative  size  to  be  given  to  the 
two  ends  of  a  room,  drawn  in  parallel  perspective,  or 
to  the  two  ends  of  a  street,  depends  thus  entirely  on 
the  position  of  the  spectator,  and  not  at  all  on  the  real 
length  of  the  side,  it  follows  that  a  long  room  seen  from 
one  point  may  be  drawn  to  look  just  like  a  short  room 
seen  from  a  nearer  point,  and  that  there  is  no  knowing 
which  is  which.  This  is  illustrated  in  Fio-.  21,  where 
the  front  half  of  the  room  in  the  upper  figure,      Fig.  21. 

a,  as  seen  from  S2  is  just  the  shape  of  the  whole  room  in 
the  lower  figure,  &,  as  seen  from  Si.  The  same  draw- 
ing also,  if  regarded  from  different  distances,  may  give 
the  impression  of  a  long  street  or  room,  or  of  a  short 
one. 

147.  In  sketching  interiors  on  the  spot,  the  point  of 
view  is  generally  excessively  near.  Such  sketches, 
when  viewed  from  a  greater  distance,  generally  give  too 
great  an  impression  of  length,  and  often  have  to  be  re- 


104  MODERN   PERSPECTIVE. 

drawn,  so  as  to  show  the  room  as  it  would  look  from  a 
point  which  really  is  outside  of  it. 

148.  Parallel  perspective  is  not  often  used  for  a  single 

Practical       objcct,  iuasmuch  as  in  order  to  show  a  sec- 
Limitations  .  .     . 
in  the  use  of  Qud  sidc,  at  rl^ht  ano'les  to  the  picture,  it  is 

Parallel  Per-  ^  o  o  a  ' 

spective.  nccessary  to  set  it  a  good  way  from  the  centre. 
Fig.  23  shows  the  top  of  a  chimney,  the  end  rig.  23. 
of  which  is  parallel  to  the  picture,  while  the  long  side  is 
perpendicular  to  it.  The  horizontal  lines  of  the  brick- 
work on  this  side  are  directed  to  V^,  those  on  the  end  are 
parallel  to  the  Horizon,  while  the  vanishing  lines  of  45°, 
as  seen  on  the  perspective  plan  below,  are  directed  to 
V^  and  V^. 

149.  It  is  to  be  noticed  that  all  the  surfaces  on  the 
end  of  the  chimney  are  drawn  of  their  true  shape  and 
proportion,  as  if  seen  in  elevation.  Still  the  whole  end 
of  the  chimney  is  not  drawn  in  elevation,  the  relations  of 
the  several  parts  being  changed  and  the  symmetry  of  the 
whole  disturbed,  since  the  nearer  surfaces  are  set  fur- 
ther to  the  right  and  higher  up,  and  since  something  is 
seen  of  the  horizontal  surfaces  that  separate  them,  which 
in  the  elevation  just  below,  drawn  in  orthographic  pro- 
jection, are  not  seen  at  all.  The  chimney  certainly  looks 
very  ill  drawn,  and  it  is  not  easy,  even  by  keeping  the 
eye  sedulously  at  the  station  point,  opposite  Y^,  to  make 
it  look  quite  right. 

150.  The  use  of  parallel  perspective  is  accordingly 
pretty  much  confined  to  cases  where  two  objects  are  to 
be  shown,  one  on  the  right  and  one  on  the  left,  as  in 


PARALLEL   PERSPECTIVE.  105 

street  views,  or  interiors.  In  tliese  cases  the  eye  natu- 
rally takes  a  central  position,  opposite  the  middle  of  the 
street  or  the  middle  of  the  room  represented.  It  is  not 
necessary,  of  course,  that  the  Axis  should  be  exactly 
in  the  middle,  and  it  is  generally  taken  near  one  side, 
so  as  to  show  as  much  as  possible  of  the  other,  and  thus 
prevent  an  absolute  symmetry. 

Notation.  Of  tlie  lines  which  are  parallel  to  the  picture,  the  ver- 
tical ones,  going  to  the  zenith,  are  lettered  Z  ;  the  horizontal  ones  are 
lettered  K,  instead  of  H,  which  would  be  confusing  ;  those  which  in- 
cline down  to  the  right  are  lettered  D,  for  dexter,  as  in  heraldry,  and 
those  which  incline  down  to  the  left  S,  for  sinister.  Those  which  slope 
directly  backward,  up  or  down,  are  lettered  A  and  A',  ior  altitude,  as 
in  astronomy.  The  planes  normal  to  the  picture  are  lettered,  accord- 
ingly, C  Z,  C  K  (  =  R  L),  CD,  and  C  S,  and  the  horizons  H  C  Z,  H  C  K 
(  =  H  R  L),  H  C  D,  and  H  C  S,  each  plane  being  designated  by  its  hori- 
zontal  element  and  its  line  of  greatest  inclination,  as  usual. 


CHAPTER  VIII. 

OBLIQUE,   OR    THREE-POINT   PERSPECTIVE. 

151.  The  last  chapter  discussed  the  phenomena  of 

Parallel  Perspective,  in  which,  of  the  three 
Poiii,V'oj;      sets  of  lines  that  define  a  rectans^ular  object 

Parallel  Per-  °  '^         ' 

spective.  ^^^Q  g^j.g  pr^i^r^iigi  to  the  picturc  and  have  their 
vanishing  points  accordingly  at  an  infinite  distance ;  the 
third  alone  has  its  vanishing  point  in  the  plane  of 
the  picture.  This  may  be  called,  accordingly,  "  One- 
Point  Perspective,"  since  it  employs  only  one  vanish- 
ing point,  V^  at  V^. 

152.  In  the  previous  chapters  only  one  of  the  princi- 
..^^^  pal  sets  of  lines,  namely,  the  vertical  lines, 
An^uiar^Per-  ^crc  parallel  to  the  picture,  both  sets  of  hori- 
spective.  2ontal  llucs  being  inclined  to  it  at  an  angle, 
one  to  the  right  and  one  to  the  left.  This  may  accord- 
ingly be  called  "  Angular  "  or  "  Two-Point  Perspective,", 
two  vanishing  points  being  employed,  Y^  and  V^. 

153.  We  now  come  to  a  third  case,  that  in  which  all 
"Three-  tlircc  of  the  principal  lines  of  a  rectangular 
obHqul  Per-  objcct  are  inclined  to  the  picture,  the  object 

presenting  towards  the  eye  a  solid  corner. 
In  this  case,  all  three  vanishing  points  are  employed, 
and   the   drawing  may   be  said   to    be   made  in  "  Ob- 


OBLIQUE,  Oil   THliEE-POINT   rEKSPECTlVE.  107 

lique,"  or  "  Three-Point  Perspective."     Plate  VII.  illus- 
trates this  case,  Figs.  24,  25,  and  26  present-  Piatevii. 
ing  examples  in  which,  though  the  object  is  I'^is^- 24, 25, 
vertical,  the  plane  of  the  picture  is  inclined ;  while  in 
Fig.  27  the  picture  is  vertical,  as  usual,  but     Fig.  27. 
the  cubical  block  on  the  floor,  the  two  covers  of  the  box 
in  the  foreground,  and  the  chair,  are  all  tipped  so  that  aU 
their  edges  are  inclined  to  the  picture.     They  are  accord- 
ingly drawn  in  Three-Point  Perspective. 

154.  Fig.  24  shows  a  post  at  the  corner  of  a  fence  as 
it  appears  when  one  looks  down  upon  it,  the     Fig.  24. 
plane  of  the  picture  being  inclined  backwards  at  the  top. 
Fig.  25  is  a  drawing  of  the  tower  of  old  Trin-     Fig.  25. 
ity  Church,  in  Boston,  which  was  destroyed  by  the  fire 
in  November,  1872,  as  it  appeared  when  one  was  looking 
up  at  it,  the  top  of  thje  picture  being  inclined  forward. 
Fig.  26  is  a  similar  view  of  the  tower  and  spire     Fig.  26. 
of  Salisbury  Cathedral,  taken  from  a  photograph.     The 
vanishing  points  in  Fig.  24  are  at  Vj,  V2,  and  Vg ;  those 
of  Fig.  25  at  V4,  V5,  and  Vg ;  and  those  of  Fig.  26  are 
not  shown,  but  can  easily  be  found. 

155.  To  make  these  drawings  look  naturaly  the  paper 
should  be  held  at  an  angle,  below  the  eye  for  the  first 
and  above  the  eye  for  the  other  two.  The  vanishing 
points  of  the  vertical  lines,  V3  and  Vg,  should  be  just 
above  or  below  the  eye. 

156.  Fig.  27  illustrates  all  three  kinds  of  perspective, 
the  room  being  drawn  in  parallel  perspective,     Fig.  27. 
with  only  one  vanishing  point,  at  V^ ;  the  bookcase  and 
the   box  in  angular   perspective,  with   two   vanishing 


108  MODERN   PERSPECTIVE. 

points,  at  V^  and  Y^ ;  and  the  lids  of  the  box,  with  the 
chair  and  the  cubical  block,  in  Three-Point  Perspective, 
with  vanishing  points  at  V^,  V^,  and  V^.  The  three 
sets  of  planes,  as  marked  on  the  cube,  are  of  course  L  M, 
L  0,  and  M  0  ;  and  their  traces,  H  L  M,  H  L  O,  and 
H  M  0,  form  a  triangle  lying  between  the  three  vanish- 
ing points. 

157.  A  plane  of  measures  is  supposed  to  pass  through 
the  nearest  corner  of  the  cube.  In  this  lie  three  lines 
of  measures,  I  m,  I  o,  and  m  o,  parallel  to  the  three 
traces  (79).  On  each  of  these  lines  the  real  length  of 
the  edge  of  the  cube  is  measured  off,  giving  the  six 
points  /,  /,  7^1,  m,  o,  o.  These  dimensions  are  trans- 
ferred to  the  three  edges  of  the  cube,  L,  M,  and  0, 
by  means  of  the  points  of  distance  D^,  D^^  and  D^, 
which  indicate  the  distance  of  each  of  these  vanish- 
ing points  from  the  station  point,  S,  in  front  of  the 
picture,  opposite  C.  Each  of  these  points  of  distance 
occurs  twice,  once  on  each  of  the  horizons  that  meet 
at  its  vanishing  point  (120). 

158.  In  the  same  way  the  width  of  the  box  is  laid  off 
on  a  vertical  line  passing  through  its  front  corner,  and 
transferred  to  its  right-hand  edge  by  means  of  D^,  in 
the  trace  HMO  or  HE  Z.  Half  the  width  of  the  box, 
which  is  the  width  of  each  half  of  the  cover,  is  trans- 
ferred to  the  inclined  lines  of  the  cover,  directed  to  V^ 
and  to  V^,  by  means  of  the  points  of  distance,  D^  and  D^, 
on  HMO. 

159.  Tills  is  all  exactly  in  accordance  with  what  has 


OBLIQUE,  OR   THREE-POINT   PERSPECTIVE.  109 

been  done  in  previous  cases,  and  involves  no  new  prin- 
ciple. The  only  new  question  which  oblique  The  problem 
perspective  presents  relates  to  the  position  of  point  Ind^S 

•  a         o      1  -trn       n    1  *^^  Centre  of 

the  station  point,  b,  oi  the  centre,  V*-,  oi  the  the  picture, 

the  three 

various  points   of  distance,  and  of  the  three  vanishing 

'■  Points  being 

vanishing  points.  Their  relations  are  ob-  ^'^^°" 
viously  much  more  strictly  defined  than  in  the  previous 
cases.  For,  in  One-Point  Perspective,  the  vanishing 
point,  V^,  being  given,  the  station  point  may  be  any- 
where upon  the  Axis,  the  line  passing  through  Y^  in  a 
direction  perpendicular  to  the  plane  of  the  picture. 
In  Two-Point  Perspective,  the  vanishing  points  V^  and 
V^  being  given,  the  station  point,  S,  must  be  somewhere 
on  the  circumference  of  a  semicircle,  whose  diameter  lies 
between  those  points,  and  which  is  itself  in  a  horizon- 
tal plane  perpendicular  to  the  plane  of  the  picture.  But 
any  point  in  this  semicircle  will  do.  In  Three-Point 
Perspective,  it  must  in  like  manner  lie  somewhere  in 
the  circumference  of  each  of  three  semicircles,  whose 
diameters  are  the  three  sides  of  the  triangle  formed  by 
the  three  vanishing  points.  For,  since  the  tliree  edges 
of  the  rectangular  object  form  right  angles  with  each 
other,  the  lines  drawn  from  the  eye  to  the  vanishing 
points  parallel  to  those  edges  must  also  be  at  right 
angles  with  each  other.  These  three  lines,  in  fact,  to- 
gether with  the  three  horizons,  form  a  triangular  pyra- 
mid, the  vertex  of  which,  at  S,  is  composed  of  three 
right  angles.  This  pyramid  is  of  the  same  shape,  obvi- 
ously, as  the  small  triangular  pyramid  that  would  be 
formed  by  cutting  across  the  corner  of  the  object  rep- 


110  MODERN   PERSPECTIVE. 

resented  with  a  plane  parallel  to  the  plane  of  the  pic- 
ture. The  lines  of  intersection  in  each  plane  would  of 
course  be  parallel  to  the  horizon  of  that  plane  (78).  The 
Fig.  27.  corner  of  the  cube  in  Fig.  27  is  represented  as 
cut  across  in  this  way. 

160.  Now  it  is  obvious  that  only  one  such  pyramid 
can  be  constructed  upon  a  given  triangle  as  a  base ;  that 
is  to  say,  given  the  three  vanishing  points,  the  position 
of  the  point  S  is  absolutely  fixed;  there  is  only  one 
point  at  which  the  eye  can  be  placed  and  find  each  pair 

Fig.  28.  of  vanishing  points  90°  apart.  Fig.  28  illus- 
trates this,  the  semicircles  that  contain  the  three  right 
angles  being  foreshortened  into  ellipses. 

161.  Another  way  of  regarding  the  problem  is  to  con- 
sider that,  since  the  plane  in  which  each  semicircle  lies 
is  not  perpendicular  to  the  picture,  but  is  inclined  to  it 
at  an  unknown  angle,  the  position  of  the  station  point 
is  really  limited  only  by  the  condition  that  it  must  lie 
somewhere  in  the  surface  of  a  hemisphere  of  which  the 
given  horizon  is  a  diameter.  As  this  is  true  for  each  of  the 
three  horizons,  the  station  point  must  be  a  point  common 
to  the  three  hemispheres.  Now  three  hemispheres,  whose 
diameters  form  a  triangle,  can  have  but  a  single  point  in 
common.  Two  of  them  will  intersect  each  other  in  a 
semicircle  perpendicular  to  the  plane  of  the  triangle, 
and  the  point  where  this  semicircle  is  cut  by  the  third 

Fig.  29.  hemisphere  will  be  the  point  in  question.  Fig. 
29  illustrates  this  view  of  the  subject. 

162.  It  is  plain  from  an  inspection  of  the  figure,  and 
of  the  little  figure  alongside,  that  the  small  semicircles  in 


OBLIQUE,    OR   THREE-POINT    PERSPECTIVE.  Ill 

which  these  hemispheres  intersect  will  be  projected  as 
straight  lines  at  right  angles  to  the  lines  connecting 
their  centres.     But  as   the   lines    connectinsr    m  «  ^„, 

o       io  nnd  the 

the  three  centres  are  obviously  parallel  to  the  ^^''*'"®' 
three  diameters,  it  follows  that  the  three  straight  lines 
in  which  these  three  semicircles  are  projected,  and 
which  meet  in  the  point  V^,  the  projection  of  the  apex  of 
the  pyramid,  are  drawn  from  the  corners  of  the  base  per- 
pendicular to  the  opposite  sides.  This  affords  an  easy 
method  of  determining  the  point  V^. 

163.  This  proposition,  that  the  projection  of  each 
edge  of  the  pyramid  is  perpendicular  to  the  opposite 
side  of  the  base,  is,  in  fact,  merely  an  illustration  of  the 
familiar  proposition  that  if  a  line  is  normal  to  a  plane, 
its  projection  upon  a  second  plane  intersecting  the 
first  is  perpendicular  to  the  line  of  intersection.  Each 
edge  of  the  pyramid  is  obviously  normal  to  the  oppo- 
site face  of  the  pyramid,  and  its  projection  upon  the 
base  must  accordingly  be  perpendicular  to  the  oppo- 
site side  of  the  base,  where  the  face  of  the  pyramid 
cuts  it. 

164.  Note.  This  is  not  the  place  to  demonstrate  the  proposition,  of 
which  the  demonstration  is  to  be  sought  in  the  treatises  on  plane  geo- 
metry, that  perpendiculars  let  fall  from  the  vertices  of  a  triangle  upon 
the  opposite  sides  will  meet  at  a  point.  But  it  is  worth  while,  perhaps, 
to  observe  that  this  point  of  symmetry  within  the  triangle  is  only  one 
of  four  such  points,  the  others  being  (a)  the  centre  of  the  inscribed 
circle,  {h)  the  centre  of  the  circumscribed  circle,  and  (c)  the  centre  of 

gravity.     Fig.  30  a,  h,  c,  d,  exhibits  a  comparative  view  of 

Fig.  oO. 
these  four  pomts. 

165.   The  point  V*"  being  thus  ascertained,  it  only  re- 


112  MODERN   PERSPECTIVE. 

mains  to  determine  the  height  of  the  pyramid,  that  is, 
the  distance  of  the  station  point,  S,  in  front  of  the  pic- 
m  .  o  .X.    '  ture,  and  the  lenoth  of  the  three  edf?es  of  the 

To  find  the  '  °  => 

Station  Point,  pyj-^nud,  that  is  to  say,  tlie  distance  of  the 
three  vanishing  points   from  the  station  point. 

Fig.  31  shows  how  these  distances  may  be  determined. 
A  plane  perpendicular  to  the  picture  is  passed  through 
either  edge  of  the  pyramid.  Its  intersection  with  the 
opposite  face  and  with  the  plane  of  the  picture,  or  base 
of  the  pyramid,  will  form  a  right-angled  triangle.  This 
triangle,  when  revolved  about  its  hypothenuse  into  the 
plane  of  the  picture,  will  give  V  S,  the  length  of  the 
edge  in  question,  and  the  height  of  the  pyramid,  or  dis- 
tance of  the  eye  from  the  picture,  V^  S.    This 

Fig.  27. 

operation  is  repeated  on  a  larger  scale  in  Fig. 
27,  giving  Sj. 

166.  Fig.  28  shows  how  the  distance  of  the  eye  from 

two  vanishing  points,  that  is  to  say,  the  length 
of  the  two  edges  of  the  pyramid,  can  be  found 
at  once  by  revolving  one  of  its  triangular  faces  into  the 
plane  of  the  picture.  Each  semi-ellipse  becomes  a 
semicircle,  on  the  circumference  of  which  is  found  the 
station  point  in  its  revolved  position  at  D,  and  D  A  and 
D  B  are  the  length  of  two  of  the  edges. 

167.  Fig.  32  exhibits  the  curious  geometrical  rela- 
Fig.  32.       tions  that  result  from  the  application  of  this 

Geometrical  p^ocess  to  all  thrcc  faccs  at  once.     It  will  be 
noticed   that   the  two  semicircles  that   start 
from  each  vanishing  point  meet  and  intersect  on  the  op- 
posite horizon,  just  at  tlie  })oint  where  tlie  perpendicular 


OBLIQUE,    OR    THREE-rOIXT   PERSPECTIVE.  113 

drawn  from  the  vanishing  point  in  qnestion  tliroiigh  the 
centre,  V^  strikes  it.  If  now,  fi-oni  each  vanisliing  point 
as  a  centre  an  arc  be  drawn  with  a  radins  equal  to  the 
distance  of  that  vanishing  point  from  tlie  station  point, 
eacli  arc  will  be  the  locics  of  its  point  of  distance,  and 
the  intersection  of  these  arcs  with  the  horizons  will  give 
the  six  points  of  distance  sought.  Moreover,  not  only 
will  each  of  these  arcs  pass  through  two  out  of  the  three 
revolved  positions  of  S,  but  its  points  of  intersection 
with  the  other  two  arcs  will  lie  in  tlie  perpendiculars 
let  fall  from  the  other  two  vanishing  points  upon  the 
opposite  traces. 

168.  This  last  observation  gives  the  means  of  deter- 
minino-  all  six  points  of  distance  by  revolviui*'  to  find  the 

°  -^  _  '^     _  ^     six  Points  of 

into  the  plane  of  the  picture  only  a  single  one  Distance, 
of  tlie  faces  of  the  pyramid,  as  is  illustrated  in  Fig.  33. 
If  the  triangle  V^^  S  V^,  right-angled  at  S,  is  revolved 
around  V^  V^,  S  will  fall  at  D,  and  the  points  of  dis- 
tance D^  and  D^,  two  of  each,  are  easily  determined, 
V^  D^^  being  equal  to  V^  D,  and  V^  D^  to  V^  D. 
But  the  locus  of  D^  passes  through  the  point  where  the 
arc  D*^DD^  cuts  V^V^,  and  also  through  the  point 
where  D^  D  D^  cuts  V^^V«.  \^D^  then,  is  easily  deter- 
mined, and  the  two  points  D^  ascertained  without  fur- 
ther labor. 

The  several  points  of  distance  in  Fig.  27,  to  which 
Fig.  33  is  similar,  are  obtained  in  this  way.  Fig.  33. 

D,  which  is  D^,  since  DV^  is  obviously  equal  to 
SV^,  enables  us  to  determine  another  D^  just  be- 
low Y^. 


114  MODERN  PERSPECTIVE. 

169.  The  phenomena  of  intersecting  planes,  with 
the  vanishing  points  of  their  lines  of  intersection  at 
the  intersection  of  their  horizons,  are  the  same  in  Three- 
Point  as  in  Two-Point  or  in  One-Point  Perspective,  and 
are  again  and  again  illustrated  in  the  plate. 


CHAPTER   IX. 

THE   PERSPECTIVE   OF   SHADOWS. 

170.  The  rays  of  the  sun,  being  practically  parallel, 

constitute  a  single  system  of  parallel  lines,  The  pheno- 
mena of 
with   the    same   two   vanishing   points,   180°  shadows. 

apart  (5).  Both  these  vanishing  points  may  be  found 
by  looking  in  the  direction  followed  by  the  rays.  If  one 
looks  up  in  the  direction  of  the  rays,  he  of  course  sees 
the  sun  itself  If  he  looks  down,  away  from  the  sun, 
he  sees  the  shadow  of  his  own  head.  Of  the  two  van- 
ishing points  of  the  system,  then,  one  is  in  the  sun  itself, 
and  the  other,  just  opposite,  is  in  the  shadow  of  the 
spectator's  head,  and  is,  of  course,  as  far  below  the  hori- 
zon as  the  sun  is  above  it. 

It  sometimes  happens  in  photographs  that  the  shadow 
of  the  camera  is  seen  in  the  foreground,  at  the  van- 
ishing point  of  shadows. 

171.  If  the  sun  is  in  front  of  the  spectator,  it  is  the 
first  of  these  vanishing  points,  that  in  the  sun   The  position 
itself,  which  is  behind  the  plane  of  the  picture,   ^f*^^^"'^- 
as  in  Fig.  34,  and  the  vanishing  point  of  the     ^'^s-^^- 
sun's  rays,  V^,  is  above  the  horizon.     If  the  sun  is  be- 
hind the  spectator,  as  in  Fi^.  35,  the  other 

.   ,  .  o  '  Fig  35_ 

vanishing  point  is  in  the  plane  of  the  picture. 


116  MODERN   PERSPECTIVE. 

and  Y^,  which  we  now  call  the  vanishing  point  of  shad- 
ows, since  every  point  throws  its  shadow  towards  it, 
is  below  the  horizon.  It  appears  in  the  figure  in  the 
extreme  right-hand  lower  corner  of  the  plate,  beyond 

Fig.  36.     If  the  sun  is  just  in  the  plane  of  the 
Fig.  36.  .  ^  '^  ^ 

picture,  neither  in  front  of  the  spectator  nor 

behind  him,  but  on  one  side  and  above,  tlie  light  falls 
parallel  with  the  plane  of  the  picture,  and  both  vanish- 
ing points  are  at  an  infinite  distance  upon  that  plane. 
This  is  illustrated  in  Fig.  36. 

172.  The  shadow  of  every  point  is  accordingly  a  line 
The  shadow  Proceeding  from  that  point,  through  the  air, 
of  a  point.  towards  the  vanishing  point  of  shadows.  This 
line  is  generally  invisible,  the  air  being  generally  trans- 
parent ;  but  when  the  air  is  loaded  with  dust  or  mois- 
ture this  line  of  shadow  becomes  visible,  as  is  often 
witnessed  at  sunset,  when  the  shadows  of  clouds  near 
the  western  horizon  are  thrown  across  the  sky  in  parallel 
lines,  —  lines  which  seem  to  converge  towards  the  sun 
in  the  west,  and  in  the  east  to  converge  towards  the 
vanishing  point  of  shadows  opposite  the  sun.  If  this 
invisible  shadow  of  a  point  strikes  any  solid  object,  it 
becomes  visible  as  a  point  of  shadow  on  its  surface.  The 
invisible  shadow  of  a  point,  then,  is  a  line  in  space ;  the 
visible  shadow  of  a  point  is  a  point  situated  where  the 
line  of  invisible  shadow  pierces  any  intercepting  surface 
upon  which  the  shadow  may  fall. 

173.  In  like  manner,  the  invisible  shadow  of  a  line  is 
The  shadow    ^  surfacc  iu  spacc,  and  the  visible  shadow  of 

the  line  is  a  line,  being  the  line  in  which  this 


THE   PERSPECTIVE   OF   SHADOWS.  117 

surface  intersects  the  surface  upon  which  the  shadow- 
falls.  If  the  line  that  casts  the  shadow  is  curved,  the 
invisible  shadow  is  cylindrical ;  if  it  is  a  straight  line 
that  casts  the  shadow,  the  shadow  in  space  is  a  plane ; 
and  if  the  surface  that  receives  it  is  also  a  plane,  the  line 
of  visible  shadow  is  a  straight  line,  being  the  line  of  in- 
tersection of  two  planes. 

In  Fig.  37  both  the  visible  and  the  invisible  shadows 
of  a  line  are  represented.  The  sun  is  supposed 
to  be  behind  the  spectator,  in  such  a  position 
that  the  shadow  of  the  spectator's  head  is  thrown  upon 
the  ground  within  tlie  limits  of  the  picture  at  V^.  The 
rays  of  light  and  the  shadow  of  every  point  in  the  line 
are  directed  towards  this  point.  The  shadow  in  space  is 
seen  to  be  a  plane,  and  the  shadow  on  the  ground  and 
steps  is  seen  to  be  the  intersection  of  this  plane  with 
the  several  planes  wliich  it  encounters. 

174.  If  a  solid  body  casts  a  shadow,  the  invisible 
shadow,  passing  downward  through  the  air  The  shadow 
away  from  the  sun,  is  a  solid  cylinder,  or  solid  body, 
prism,  according  as  the  line  upon  the  body  that  casts  tlie 
shadow  is  a  curved  line  or  rectilinear.  Tliis  line  is  ob- 
viously the  line  upon  the  surface  of  the  body  which 
separates  the  side  towards  the  sun,  which  is  in  light, 
from  the  shady  side.  This  line  is  called  the  dividing 
line  of  light  and  shade.  The  visible  shadow  of  the  solid 
object,  seen  upon  any  other  object,  is  a  surface,  the  shape 
of  wdiich  is  determined  by  the  shadow  cast  by  the  line 
of  light  and  shade. 

To  find  the  shadow  of  a  solid  body,  then,  is  the  same 


118  MODERN    PERSPECTIVE. 

thing  as  to  find  the  shadow  of  a  line,  namely,  the  shadow 
of  its  dividing  line  of  light  and  shade. 

175.  In  finding  the  shadow  of  a  point,  also,  the  only 
practicable  way  is  first  to  find  the  shadow  of  some  line 
that  passes  through  the  point,  and  then  to  find  in  this 
line  of  shadow  the  point  of  sliadow  that  corresponds  to 
it.  This  point  is  easily  found  by  drawing  a  line,  repre- 
senting the  invisible  shadow  of  the  point,  through  the 
air,  from  the  perspective  of  the  point  to  the  vanishing 
point  of  shadows,  V^.  Its  point  of  intersection  with  the 
visible  shadow  of  the  auxiliary  line  is  the  shadow  of  the 
point  in  question. 

Thus  in  Fig.  37  the  point  A  has  its  shadow  at  a;  and, 
conversely,  the  shadow  at  h,  at  the  bottom  of 

Fig.  37.  "^ . 

the  steps,  is  cast  by  the  point  B.  This  shows 
just  how  much  of  the  stick  throws  its  shadow  on  the 

ground.     In  Fig.   35  the  shadows  of  all  the 
Fig.  35.        ^  .  .      ^ 

prnicipal  points,  such  as  the  top  of  the  sign- 
post, or  of  the  peak  of  the  gable,  are  found  by  drawing- 
lines  to  V®,  and  marking  their  points  of  intersection 
with  the  shadows  of  the  lines  in  which  these  principal 
points  lie. 

176.  The  whole  problem  of  shadows  thus  resolves 

itself  into  the  problem  of  finding  the  shadow 

The  visible  .  ^  ° 

shadow  of  a    of  a  line;  and  as  in  this  paper  we  shall  con- 

hne :   the  in-  '  r    r 

thrpiane  of  ^Idcr  ouly  the  case  of  straight  lines  throwing 

its  invisible      1.1      •        i       i  i  p  ^  . 

shadow  with  their  shadows  upon  plane  surfaces,  we  have  to 

which  it        do  only  with  rectilinear  shadows,  lying  where 

the  plane  of  the  invisible  shadow  of  the  line 


THE   PERSPECTIVE    OF   SHADOWS.  119 

cuts  the  plane  of  the  surface  that  receives  it.  The 
whole  question  becomes,  then,  simply  a  question  of  the 
intersection  of  planes. 

177.  Now,  the  line  of  intersection  of  two  planes,  as 
we  have  seen  in  the  case  of  two  intersecting  roof  planes, 
has  its  vanishing  point  at  the  point  of  intersection  of 
the  horizons  of  those  planes  (34).  Hence  the  (visible) 
shadow  of  a  line  upon  a  given  plane  will  have  its  van- 
ishing point  at  the  intersection  of  the  horizon  of  that 
plane  with  the  horizon  of  the  plane  of  the  (invisible) 
shadow  of  the  line.  And  since,  if  any  plane  is  given  in 
perspective,  its  horizon  is  already  known,  the  only  thing 
that  remains  to  be  done  is  to  find  the  horizon  of  the 
plane  of  the  shadow.  The  direction  of  the  line  and  the 
direction  of  the  light  are,  of  course,  also  given ;  that  is 
to  say,  their  vanishing  points  also  are  known. 

178.  But  these  two  vanishing  points  being  known, 
the  horizon  of  the  invisible  plane  of  the  shadow  . 

i  The  horizon 

is  easy  to  ascertain.    For  the  horizon  of  a  plane  ^J  JJ^  ^^l 
passes   through  the  vanishing  points   of   any 
two  lines  that  lie  in  it  (13,  II.).     Now,  as  may  be  seen 
in  Figr.  37,  the  line  that  casts  it  lies  in  the 

...  Fig.  37. 

plane  of  the  shadow,  and  so  does  the  invisible 
shadow  of  any  point  in  that  line.  The  horizon  of  the 
plane  of  the  shadow  accordingly  passes  through  the  van- 
ishing point  of  the  line  that  casts  it,  and  through  the 
vanishing  point  of  shadows  Y^  and  may  be  found  at 
once  by  drawing  a  line  through  them. 

179.  Thus  in  Fig.  34  the  horizon  of  the  shadows  of 
the  right-hand  horizontal  lines  E,  whose  vanishing  point 


120  MODEEN    PERSPECTIVE. 

is  V^,  is  tlie  line  H  S  E,  the  horizon  of  the  shadow  of  R, 
ioinina;  V^  and  V®.     In  the  same  way,  if  we 

Fig.  34.  The    'JO  -^  ' 

sun  in  front   call  the  plane  of  the  shadow  of  L,  SL;  that 

of  the  spec-  ^  '  ^ 

SdThe  of  M,  SM ;  that  of  Z,  S  Z,  etc.,  we  sliall  liave 
picture.  ^^g  horizons  of  these  planes,  H  S  L,  H  S  M, 
H  S  Z,  etc.,  connecting  Y®  with  V^  V^^  V^  etc.,  respec- 
tively. As  V^,  the  vanishing  point  of  vertical  lines,  is 
at  an  infinite  distance  in  the  zenith,  H  S  Z,  like  H  R  Z 
and  H  L  Z,  is  a  vertical  line. 

It  is  not  in  general  very  easy  to  follow  these  invisible 
planes  of  shadow  in  imagination,  and  to  understand  just 
how  they  go,  by  a  mere  inspection  of  the  figure.  But  in 
the  case  of  a  vertical  line,  such  as  that  of  the  nearest 
corner  of  the  building,  one  can  see  that  the  plane  of  the 
shadow  must  be  a  vertical  plane  nearly  parallel  with  the 
right-hand  side  of  the  house,  but  not  quite  so,  being  at 
a  less  angle  with  the  plane  of  the  picture.  It  seems 
reasonable,  then,  to  find  its  horizon  H  S  Z  parallel  with 
H  R  Z  and  a  little  further  to  the  rioht. 

o 

180.    Since  all  these  planes  of  shadow  have  one  ele- 
35  Ti     ^^^^^t  parallel  to  the  light,  all  their  horizons, 
the  s^pect£     ^^  ^^  ^^^^^  ^^oth  in  Fig.  34  and  in  Fig.  35,  pass 
*°'^'  through  V^.      This    point  thus   furnishes   an 

illustration  of  the  proposition  (13,  h)  that  the  horizons 
of  all  the  systems  of  planes  that  can  be  passed  through 
a  line,  or  drawn  parallel  to  it,  in  any  direction,  pass 
througli  the  vanishing  point  of  the  system  to  which 
the  line  belongs,  and  intersect  each  other  at  that 
point. 

V^  accordingly,  resembles  the  centre  of  a  wheel,  the 


THE   PEKSPECTIVE   OF   SHADOWS.  121 

spokes  of  which  are  drawn  through  the  vanishing  points 
of  all  the  lines  in  the  picture. 

181.   The  intersection  of  the  horizons  of  these  planes 
of  shadow  with  the  horizons  of  the  different 
planes  on  which  the  shadows  fall  gives  the  Ji^J'pdnt^of 
vanishing  points  of  the  different  lines  of  visi-  the  vSe 
Lie  shadow.     Thus  in  Fig.  35  the  horizon  of 
the  shadow  of  the  sign-post  is  H  S  Z  ;  and  the 
successive  portions  of  its  shadow  which  fall  upon  the 
ground,  E  L,  upon  the  side  of  the  house,  L  Z,  and  upon 
the  roof,  LM,  are  directed  to  the  points  of  intersection 
of  H  S  Z  with  H  R  L,  or  the  Horizon,  with  H  L  Z  and 
with  H  L  M  respectively. 

Fig.  38  exhibits  these  relations  in  a  diagram.     The 
vanisliiui'-  points  of  shadows  are  marked  in 

Fig.  38. 

this  plate  by  four  letters,  thus,  V^^'^^,  which 
signifies  the  vanishing  point  of  the  shadow  of  vertical 
lines  falling  upon  the  plane  L  M.  It  hardly  needs  to  be 
pointed  out  that  all  the  different  shadows  cast  by  the 
lines  of  any  system,  on  whatever  plane  they  fall,  liave 
their  vanishing  points  on  the  horizon  of  the  shadow  of 
that  system;  and  that  all  the  shadows  that  fall  on  a 
plane,  whatever  kind  of  line  casts  them,  have  their 
vanishing  points  in  the  horizon  of  that  plane. 


182.    The  direction  of  the  lines  of  shadow  being  thus 
predetermined  by  the  determination  of  their 
vanishinor  points,  and  their  length  beinir  fixed  point  of  a 

>=^   ^  '  o  o  shadow. 

either  by  the  limits  of  the  plane  on  which  they 

fall  or  by  the  limits  of  the  length  of  the  lines  that  cast 


122  MODERN   PERSPECTIVE. 

them,  everything  is  known  about  them  except  their 
exact  position.  To  fix  this  it  is  necessary  to  know  the 
position  of  some  one  point  in  the  line  of  shadow.  This 
is  generally  given  in  the  conditions  of  the  problem.  In 
Fig.  35,  for  example,  so  much  of  the  shadow 
of  the  sign-post  as  falls  on  the  ground  is  deter- 
mined in  position  by  its  initial  point.  The  shadow  be- 
gins where  the  pole  touches  the  ground.  Thence  it 
goes  off  in  the  direction  of  its  vanishing  point,  at  the 
intersection  of  H  S  Z  with  the  horizon,  as  far  as  the 
ground  extends;  that  is,  to  the  wall  of  the  house. 
The  terminal  point  of  the  shadow  on  the  ground  is 
the  initial  p(jint  of  the  portion  that  runs  up  the  wall, 
and  so  on. 

All  the  shadows  in  Fig.  34  and  Fig.  35  are  drawn  in 
this  way,  and  illustrate  these  principles.     It 

Figs.  34, 35.      .  1  r 

IS  not  worth  while  to  take  space  to  describe  in 
detail  what  may  now  easily  be  understood  from  an  in- 
spection of  the  plate. 

183.  If  no  convenient  spot  to  begin  at  is  furnished  by 
the  conditions,  it  is  necessary  either  to  prolong  the  line 
and  extend  the  plane  until  they  meet,  in  which  case 
the  initial  point  of  the  shadow  is  the  point  at  which 
the  line  pierces  the  plane,  or  to  pass  an  auxiliary  line, 
in  any  direction  that  is  most  convenient,  through  some 
point  in  the  given  line,  and  to  find  the  point  where  it 
pierces  the  given  plane.  This  point  will  be  the  initial 
point  of  the  shadow  of  the  auxiliary  line ;  the  shadow 
of  the  point  selected  can  then  be  determined  upon  it, 
and  the  shadow  of  the  given  line  drawn  through  that 


THE   PEKSPECTIVE   OF   SHADOWS.  128 

point  of  shadow.  The  auxiliary  line  must  always 
be  employed  when  the  line  that  casts  the  shadow  is 
parallel,  or  nearly  parallel,  to  the  ]3lane  on  which  it 
falls. 

It  is  generally  most  convenient  to  take  this  auxiliary 
line  in  a  vertical  direction.  Tliis  is  done  in  the  case  of 
the  balloon,  shown  in  Fig.  35.  Its  position  being  known, 
a  line  can  be  dropped  from  it  upon  the  plane  beneath, 
and  the  shadow  of  the  balloon  drawn,  at  the  end  of  the 
shadow  of  the  line. 

184.  Fig.  36  illustrates  the  case  in  which  the  sun  is 
neither  behind  the  spectator  nor  behind  the  p.  ^  ^^^ 
picture,  but  just  in  the  plane  of  the  picture,  phluI^oMhe 
throwing  his  rays  parallel  to  it  and  to  the  ^^^  "^^' 
plane  of  measures.  This  is  by  far  the  most  convenient 
position  for  the  sun  when  the  objects  represented  are 
drawn  in  angular  or  two-point  perspective,  as  they  gen- 
erally are.  It  is  almost  sure  to  produce  a  picturesque 
disposition  of  light  and  sliade. 

It  is  also  much  simpler  and  easier  to  work  than  either 
of  the  other  cases.  For  since  the  vanishing  point  of 
shadows  is  at  an  infinite  distance,  V®  is  entirely  off  the 
paper,  and  the  rays  of  light  cross  the  paper  at  their  real 
inclination  with  the  ground ;  and  not  only  the  lines 
of  invisible  shadow,  but  the  horizons  of  the  planes  of 
shadow,  have  the  same  inclination.  There  is  an  appar- 
ent difficulty  in  the  case  of  vertical  lines,  and  of  other 
lines  parallel  to  the  picture,  since  their  vanishing  points, 
as  well  as  the  vanishing  point  of  shadows,  are  at  an 


124  MODERN   PERSPECTIVE. 

infinite  distance,  and  it  is  impracticable  to  find  the 
horizon  of  their  shadows  by  drawing  a  line  from  one  in- 
finitely distant  point  to  another.  But  it  is  obvious  that 
these  lines  must  cast  their  shadows  in  planes  parallel 
to  the  picture.  The  sliadow  of  such  a  line  on  any 
plane,  then,  will  be  parallel  to  the  trace  of  that  plane 
(80). 

Fig.  36  furnishes  abundant  iUustration  of  this  case. 

185.  It  is  not  always  quite  obvious,  from  mere  in- 
spection of  a  drawing,  which  of  the  edges  of  a 


The  dividins 
line  of  ligl 
and  shade 


lineofiisur   solid  objcct  really  determine  the  form   of  its 


shadow  ;  which  of  its  lines  go  to  make  up  the 
dividing  line  of  light  and  shade  (174)  ;  which  of  its  sur- 
faces, that  is,  are  turned  towards  the  sun,  and  which 
are  turned  away  from  it.  It  is  not  easy  to  tell,  for  ex- 
ample, whether  the  farther  slope  of  a  roof  is  in  the  light 
or  not ;  whether  the  eaves  or  the  ridL,^e  is  casting^  a 
shadow  on  the  ground  beyond.  Conversely,  it  is  not 
always  easy  to  judge  just  where  the  sun  must  be  put  in 
order  to  produce  the  distribution  of  light  and  shade  upon 
the  different  surfaces  that  is  desired. 

These  difficulties  disajopear,  however,  if  we  consider 

that  what  we  want  to  know  is  whether  or  not 

Sunset. 

the  sun  has  set,  so  to  speak,  to  the  plane  in 
question,  and  apply  to  that  plane  the  same  test  that  we 
apply  to  the  liorizontal  plane  of  the  ground.  If  the  sun 
is  above  the  horizon,  or  the  vanishing  point  of  shadows, 
opposite  the  sun,  is  below  the  horizon,  we  know  that  the 
ground  is  in  light,  and  vice  versa.  So  of  every  other  plane : 
if  the  sun  is  above  its  horizon,  it  is  in  light ;  if  the  sun 


THE    PEKSPEUTIVE    OF    SHADOWS.  125 

lias  set  to  it,  and   the  vanishing   point  of  shadows  is 
above  its  horizon,  the  plane  is  in  shade. 

In  Fig.  35,  for  example,  V^  is  beyond  H  R  Z ;  it  is 
above  the  horizon  of  the  plane  R  Z,  the  right- 
hand  side  of  the  house.  This  side  of  the 
house  IS  accordingly  in  the  shade  ;  the  sun  has  set  to  it. 
If  V®  were  moved  to  the  other  side  of  H  R  Z,  below  this 
horizon,  that  side  of  the  house  would  obviously  be  in 
the  liglit.  So  of  L  M',  the  plane  of  the  back  of  the  roof. 
V^  is  above  H  L  M';  the  sun  has  set  to  that  plane 
also,  and  the  dividing  line  of  light  and  shade  runs  along 
the  ridge ;  it  is  the  ridge,  not  the  eaves  beyond,  that 
casts  a  shadow. 

186.  But  it  is  to  be  noticed  that  when  the  sun  set  to 
the  end  of  the  house  which  is  in  sight  it  rose  to  the 
other  end  of  the  house  which  is  parallel  to  it,  and  as 
both  these  planes  have  H  R  Z  for  their  horizon  we 
must  discriminate  between  them. 

This  we  can  do  if  we  recall  the  distinction  already 
pointed  out  between  the  surfaces  that  are  in  sight  and 
those  that  are  not :  "  A  plane  surface  upon  a  solid  object 
cannot  be  seen  unless  it  is  on  the  side  of  the  object  next 
tlie  horizon  of  the  plane"  (12);  that  is  to  say,  unless  it 
is  below  its  horizon  (38). 

187.  Bearing  this  in  mind,  we  have  the  following 
rule  for  the  illumination  of  surfaces  by  the  sun  :  — 

A  plane  surface  that  is  in  sight,  being  turned  towards 
its  horizon,  is  in  the  light  if  the  sun  is  on  the  farther  side 
of  its  horizon,  or  if  the  vanishing  point  of  shadows  is  on 
the  hither  side. 


126  MODEKN   PERSPECTIVE. 

A  plane  surface  that  is  out  of  sight  is  in  the  light  if 
the  sun  is  on  the  hither  side  of  its  horizon,  or  the  vanish- 
ing point  of  shadows  on  the  farther  side. 

188.    It  has  not  seemed  worth  while  to  encumber  the 
figures  with  constructive  lines.     It  is  for  the 

Notation. 

most  part  left  to  the  intelligence  of  the  reader 
to  trace,  point  by  point,  the  application  of  these  prin- 
ciples in  the  various  cases  they  present.  In  Fig.  35, 
however,  a  notation  has  been  used  for  the  outline  of 

some  of  the  principal  shadows  which  will  serve 

Fig.  35. 

both  to  recall  the  principle  of  their  construc- 
tion and  to  indicate  the  point  to  which  they  are  directed. 
The  expression  "  S  Z  on  E  L,"  for  instance,  indicates 
that  the  outline  to  which  it  is  attached  is  the  shadow  of 
a  vertical  line,  Z,  on  a  horizontal  plane,  EL;  "  S  N  on 
L  M,"  in  like  manner,  when  applied  to  the  shadow  upon 
the  upper  roof  of  the  iron  rod  which  supports  the  chim- 
ney, signifies  the  shadow  of  a  line  N  upon  the  plane  L  M. 
In  both  cases  the  line  of  shadow  is  a  line  of  intersection 
of  two  planes,  and  has  its  vanishing  point  at  the  inter- 
section of  their  horizons ;  in  the  former  case  at  Y^^-  ^, 
where  H  S  Z  intersects  with  H  E  L,  in  the  latter  case 
at  V^^-  ^,  where  H  S  N  meets  H  L  M. 

189.  In  a  few  cases  the  lines  of  invisible  shadow  have 

been  indicated,  converging  to  V^,  to  show  their  use  in 

determining  the  length  of  the  visible  shadow.     In  Fig. 

34,  where  this  is  done,  it  will  be  noticed  that 

Fig  34. 

the  dotted  lines  drawn  from  the  top  of  the 
posts  converge  at  the  sun,  while  their  shadows  converge 


THE   PERSPECTIVE    OF    SHADOWS.  127 

to  the  point  on  the  horizon  below  the  sun.     In  Fig.  36 

the  visible  shadows  of  vertical  lines  are  par- 
Fig.  36. 
allel  and  horizontal,   while   the  dotted   lines 

that  indicate  the  invisible  shadows  follow  the  real 
direction  of  the  light,  falling  parallel  to  the  picture, 
and  are  parallel  to  each  other,  and  also  to  all  the  horizons 
of  shadows  drawn  through  the  various  vanishing  points. 
190.  It  will  be  observed  that  wherever  a  line  is  par- 
allel to  the  plane  on  which  it  casts  its  shadow  it  is  an 
clement  of  both  systems  of  planes  •,  the  horizons  of  both 
planes  accordingly  pass  through  its  vanishing  shadows  on 

.  .  planes  paral- 

pomt  (13,  o),  which  is  their  point  of  intersec-  leitothe 

r  \       '     y>  i  ]ines  that 

tion,  and  tlie  shadow  is  parallel  to  the  line  cast  them, 
that  casts  it,  as  it  should  be,  having  the  same  vanishing 
point.  This  is  illustrated  in  Fig.  35  by  the  shadows  cast 
by  vertical  lines  upon  the  vertical  planes,  by  horizontal 
lines  upon  the  ground,  and  by  the  inclined  lines  M  upon 
the  inclined  planes  L  M. 

The  shadows  cast  by  artificial  light  are  discussed  in 
Chapter  XIV.,  where  will  also  be  found,  in  section  313, 
an  alternative  method  of  finding  certain  shadows  cast 
by  sunlight. 


CHAPTEE  X. 

THE  PERSPECTIVE  OF  REFLECTIONS. 

191.  Let  us  now  consider  how  things  look  in  a  mir- 
ror,—  whether  in  an  artificial  mirror,  or  looking-glass, 
or  in  the  natural  mirror  formed  by  the  surface  of  still 
water.  The  question  is  obviously  a  little  more  compli- 
cated than  those  we  have  been  discussing,  inasmuch  as 
a  new  element  is  introduced.  We  have  now  to  consider 
not  only  the  position  of  the  spectator  and  the  position 
of  the  picture,  and  their  relation  to  the  position  of  the 
object  reflected  and  to  the  mirror  that  reflects  it,  but 
also  the  relation  of  the  object  reflected  and  of  the  re- 
flecting surface  to  each  other.  Either  of  these  may  be 
parallel,  perpendicular,  or  inclined  to  the  others. 

Given,  the  position  of  the  spectator,  that  of  the  pic- 
ture, that  of  the  object,  and  that  of  the  mirror,  it  is 
required  to  depict  not  only  the  lines  and  surfaces  of  the 
object  itself,  with  all  their  vanishing  points  and  horizons, 
but  the  reflection  of  the  object  in  the  mirror,  and  the 
vanishing  points  and  horizons  of  the  reflection. 

192.  In  reflections,  however,  as  in  shadows,  and  as 
everywhere  in  perspective,  the  various  problems  of  the 
point,  the  line,  and  the  surface  are  all  comprised  in  the 
problem  of  the  liue  (205,  206).     How  to  draw  a  given 


THE  PERSPECTIVE  OF  KEFLECTIONS.       129 

line  through  a  given  point  is  the  only  question.  For 
the  perspective  of  a  point  can  be  got  only  by  finding  the 
perspective  of  a  line,  or  of  two  lines,  passing  through  it, 
and  a  surface  is  drawn  in  perspective  by  drawing  the 
perspective  of  the  lines  that  enclose  it. 

The  problem  of  reflections,  then,  is  this :  to  draw  the 
reflection  of  a  given  line  in  a  given  mirror.  The  position 
of  the  centre  of  the  picture,  V^,  is,  of  course,  known,  and 
the  distance  of  the  eye  at  the  station  point,  S,  in  front  of 
it ;  and  the  vanishing  point  of  the  line  and  the  horizon 
or  horizon  of  the  plane  of  the  mirror ;  wath  the  position 
of  some  point  in  the  line  and  of  some  point  in  the 
mirror. 

193.  Let  us  first  take  the  most  general  case, —  that 
of  a  line  inclined  to  the  picture,  at  any  angle,  The  general 

case :  the  re- 
taken at  random,  reflected  in  a  mirror  which  flection  of  an 

'  obhquely  in- 

is  inclined  to  the  plane  of  the  picture  and  to  Jn'^^^^^Jb"^ 
the  horizontal  plane,  in  any  position,  taken  at  cHuei^mii- 

ror. 

random. 

This  disposition  is  shown  in  Figs.  39,  40,  and  41, 
which  illustrate  tliree  successive  steps  in  the  Figs^3<j^40, 
solution  of  the  problem. 

In  each  of  these  is  shown  two  sides  of  a  room,  making 
an  angle  with  the  plane  of  the  picture,  on  one  of  which 
a  mirror  hanos  at  an  ande  with  the  wall.  The  plane  of 
this  mirror  we  will  call,  in  pursuance  of  the  system  of 
notation  adopted  in  these  papers,  the  plane  L  M',  since 
its  horizontal  element  is  obviously  parallel  to  the  edge  of 
the  floor,  whose  vanishing  point  is  V^,  and  its  line  of  steep- 


130  MODERN    PERSPECTIVE. 

est  slope  descends  to  the  right,  in  the  general  direction 
which  we  have  called  M^  V^'  will  be  below  V^,  and 
H  L  M',  the  horizon  of  the  plane  of  the  mirror,  will  pass 
through  V^'  and  V^,  as  shown.  Tlie  position  of  the  mirror 
is  fixed  by  tlmt  of  its  lower  edge  where  it  intersects  the 
plane  of  the  floor. 

The  line  whose  reflection  in  this  mirror  we  are  to  find 
slants  upwards  to  the  left  nearly  in  the  direction  we 
have  called  N ;  but  as  it  is  not  parallel  to  the  plane  L  Z, 
we  will  call  it  0,  its  direction  being  given  by  its  vanish- 
ing point  V^.  Its  position  is  fixed  by  that  of  its  nearest 
point,  whose  distance  above  a  point  on  the  floor  is 
shown. 

Let  us  also  call  the  direction  of  lines  at  right  angles 
to  the  mirror  —  that  is  to  say,  perpendicular  or  normal 
to  it  —  by  the  letter  T,  norma  being  Latin  for  T-square. 
They  will  be  parallel  to  the  axis  of  the  mirror,  and  their 
vanishing  point  will  be  V^. 

194.  It  is  obvious  from  the  inspection  of  either  figure 
that  0',  the  reflection  or  image  of  the  line,  will  look  like 
just  such  another  line  as  far  behind  the  surface  of  the 
mirror  as  the  line  itself,  or  object,  is  in  front  of  it ;  and 
that  a  line  drawn  from  any  point  in  the  object  to  the 
corresponding  point  in  the  image  —  that  is  to  say,  from 
any  point  to  the  reflection  of  that  point  —  will  be  normal, 
or  perpendicular,  to  the  mirror,  and  will  have  its  vanish- 
ing point  at  V^. 

All  these  normal  lines  together,  moreover,  make  up  a 
normal  plane,  also  at  right  angles  to  the  mirror,  which 
may  be  called  the  plane  0  T.     And  as  the  horizon  of  any 


THE  perspectivp:  of  reflections.  131 

plane  passes  through  the  vanishing  points  of  all  the 
elements  of  the  plane,  HOT,  the  horizon  of  the  normal 
plane  0  T,  passes  through  V^,  the  vanishing  point  of 
the  given  line,  and  through  V^,  the  vanishing  point  of 
normals. 

And  as,  conversely,  every  line  that  lies  in  a  plane  has 
its  vanishing  point  in  the  horizon  of  the  plane,  0',  the 
image  of  0,  must  have  its  vanishing  point  V^'  also  in 
the  horizon  HOT. 

Finally,  it  is  plain  that  the  line  I,  in  which  this  nor- 
mal plane  intersects  the  plane  of  the  mirror,  seems  to , 
lie  equidistant  between  the  given  line  0  and  its  image 
0',  and  to  bisect  the  angle  they  make  with  each  other  ; 
and  that  since  it  lies  at  the  intersection  of  these  two 
planes,  its  vanishing  point  must  be  at  V^  the  intersec- 
tion of  their  horizon. 

195.  The  problem  of  reflections  is  solved  when  the 
line  I  is  fixed  and  the  point  V^ ;  for  the  image  or  reflec- 
tion of  every  point  in  the  given  line  lies  on  a  normal 
passed  through  the  point,  at  a  distance  beyond  the  line  I 
equal  to  that  of  the  point  itself  on  the  hither  side  of  it. 
This  equal  distance  can  be  obtained,  as  show^n,  by  means 
of  the  method  of  triangles,  using  a  line  of  measures,  o  t, 
drawn  parallel  to  HOT,  and  an  auxiliary  vanishing 
point,  at  V,  as  a  point  of  proportional  measures.  By 
taking  o  t  of  any  convenient  length,  and  then  taking  t  o 
equal  to  it,  the  initial  point  of  0'  upon  the  line  T  is 
easily  ascertained.  But  for  a  complete  solution  of  the 
problem,  it  is  necessary  to  determine  also  V^',  the  vanish- 
ing point  of  the  image. 


132  MODERN   PEKSPECTIVE. 

Fig.  89  shows  how  V^,  and,  consequently,  V^  are  de- 
j,.  39  termiued ;  while  Fig.  40  and  Fig.  41  show 
fIIs.  46, 41.    jjQ^  J  •j.ggjf  ^^^  yo/  are  obtained. 


196.  The  first  problem  is  how  to  find  V^,  H  L  M 
being  given ;  that  is  to  say,  given  a  plane  or  system  of 
planes  by  its  horizon,  to  find  the  vanishing  point  of  the 
axis  of  the  system,  i.  e.,  its  vanishing  point  of  normals. 

The  solution  of  this  problem  depends  upon  the  prin- 
ciple already  illustrated  in  Fig.  29,  Plate  VII.,  that ''  if 
a  line  is  normal  to  a  plane  its  projection  upon  a  second 
plane  intersecting  the  first  is  perpendicular  to  the  line  of 
Plate  IX  intersection."  For  if  (Fig.  39)  we  pass  through 
Fig.  39.  ^Y^^  station  point,  S,  in  the  air,  at  a  distance  in 
front  of  V^,  equal  to  the  line  V^  Sj,  a  plane  parallel  to  the 
mirror,  it  will  intersect  the  plane  of  the  picture  in  the 
The  vanish-  Huc  H  L  M',  aud  if  we  pass  through  the  same 
the  lines  nor-  Doiut  a  liuc  uomial  to  that  plane  and  parallel 

mal  to  the        ^ 

mirror.  to  the  liues  nomial  to  the  mirror,  it  will 
pierce  the  plane  of  the  picture  at  V^;  for  if  from  tlie 
station  point  one  looks  in  a  direction  parallel  to  the 
lines  of  any  system  he  will  see  the  vanishing  point  of 
that  system.  Now,  since  the  projection  of  the  station 
point  on  the  plane  of  the  picture  is  at  V^,  the  projection 
of  this  line  will  be  V^  V^,  and  this  line  will  be  perpen- 
dicular to  H  L  M'  at  the  point  a.  A  line,  then,  dnavn 
through  V^  pcrioendicular  to  the  horizon  of  any  given 
plane,  luill  pass  thro^igh  the  vanishing  point  of  lines  nor- 
mal to  that  p)lane. 

197.  If  now  a  line  be  drawn  from  the  station  point 


THE    PERSPECTIVE    OF   KE  FLECTIONS.  133 

S,  to  the  point  a,  it  will  be  at  right  angles  to  the  line 
drawn  from  the  station  point  to  V^.  a  S  V^  is  accord- 
ingly a  right-angled  triangle  of  which  a  V^  is  the  hypo- 
thenuse,  and  if  this  triangle  be  revolved  into  the  plane 
of  the  picture  about  a  V^,  S  will  fall  at  S^  (V^  S^  being 
taken  equal  to  V^  Sj),  and  a  line  drawn  at  right  angles 
with  a  S2  will  give  V^  at  its  intersection  with  a  V^  pro- 
longed. 

198.  V^  being  thus  ascertained,  HOT,  the  horizon  of 
tlie  normal  plane,  is  drawn  through  V'"'  and  The  horizon 

of  the  nor- 

V^,  and  V^  is  obtained  at  its  intersection  with  mai  plane. 
HLM'  (194). 

199.  Fig.  40  shows  how  the  line  I,  at  the  intersection 
of  the  normal  plane  with  the  mirror,  is  deter-     pj^  ^q 
mined  in  position ;  its  vanishing  point  Y^  be-  The  point 
ing   already   found,   it  is    necessary    only   to  given  une 

pierces  the 

determine   one  point  in  the  line  I,  upon  the  niirror. 
surface  of  the  mirror.     The  line  can  then   be  drawn 
through  this  point  and  the  point  V^. 

But  it  is  plain,  from  an  inspection  of  Fig.  39,  that  if 
the  line  0  were  prolonged  until  it  touched  the  mirror 
its  reflection  0'  and  the  line  I  lying  between  them  would 
be  prolonged  also,  and  that  all  three  lines  would  meet 
at  the  point  where  0  pierced  the  surface.  This  point, 
then,  would  be  a  point  of  the  line  I  such  as  we  are 
seeking. 

The  problem  resolves  itself,  then,  into  that  of  finding 
where  a  line  pierces  a  plane,  both  being  given  in  per- 
spective. 


134  MODERN   PERSPECTIVE. 

200.  To  find  where  in  the  figure  (Fig.  40)  the  line  0 
pierces  the  plane  of  the  mirror  L  M'  we  pass  a  vertical 
plane,  O  Z,  through  the  line  0  ;  its  horizon  is  H  0  Z.  The 
dotted  line  in  which  this  plane  intersects  the  floor  has 
its  vanishing  point  at  the  intersection  of  the  Horizon 
H  Pt  L,  the  horizon  of  tlie  plane  of  the  floor,  with  H  0  Z, 
the  horizon  of  this  vertical  plane  (13,  III.).  The  point 
e  is  a  point  common  to  the  plane  LM'  and  to  the 
plane  0  Z ;  that  is  to  say,  it  is  one  point  of  the  line  in 
which  the  vertical  plane  through  0  intersects  the  surface 
of  the  mirror,  the  point  /,  where  the  traces  of  these  planes 
intersect,  being  tlie  vanishing  point  of  this  line  of  inter- 
section. The  line  fe  prolonged  is  then  this  line  of 
intersection  itself,  and  the  point  g,  where  the  line  0 
prolonged  meets  the  line  /e,  is  the  point  where  it 
pierces  tlie  mirror, 

201.  But  the  point  g,  being  also  a  point  of  the  line  I 
prolonged,  a  line  drawn  through  g  and  the  vanishing 
point  V^  gives  the  indefinite  line  I  which  we  are  seek- 
inof.  Normals  drawn  throuojh  the  extremities  of  the  line 
0  to  V^  cut  off  from  the  indefinite  line  I  the  finite  por- 
tion required.     This  is  shown  in  Fig.  41. 

202.  Fig.  41  also  shows  how  V^',  the  vanishing  point  of 
Fig.  41.  The  the  reflection  of  a  given  line,  may  be  obtained, 
pofnt  of°the    V',  V^  and  I  having  already  been  determined, 

and  0  and  V^  being  given. 
It  is  plain  that  0, 1,  0',  and  T,  in  Fig.  39,  are  all  in 
the  same  normal  plane,  and  that  if  from  the  position  of 
the  eye  at  the  station  point,  S,  in  the  air,  in  front  of  V^, 


THE  PERSPECTIVE  OF  REFLECTIONS.       135 

lines  are  drawn  to  the  vanishing  points  Y^,  V^,  Y^',  and 
V^,  these  lines  will  lie  in  the  plane  which,  passing 
tlirough  the  eye,  intersects  the  plane  of  the  picture  in  the 
horizon  HOT.  The  lines  S  Y^  S  Y^,  S  Y^  and  S  Y^' 
will  all  lie  in  the  same  plane,  and  will  be  parallel  respec- 
tively to  T,  O,  I,  and  0',  and  will  make  tlie  same  angles 
one  with  another.  But  since  I  bisects  the  angle  made  by 
0  and  0'  (194)  so  must  S  Y^  bisect  the  angle  at  S,  made 
by  S  Y^  and  S  \^ '.  If,  then,  in  Fig.  41  we  revolve  the 
plane  triangle  Y*^,  S,  Y^  right-angled  at  S,  around  its 
hypothenuse,  into  the  plane  of  the  picture,  the  horizon 
HOT  and  all  the  vanishing  points  upon  it  will  remain 
where  they  are,  and  S  will  fall  at  S4.  For  the  real  dis- 
tance from  the  station  point  to  the  horizon  H  0  T  at  6  is 
6  S3,  the  hypothenuse  of  the  right-angled  triangle  of  which 
Y^^  is  the  base  and  Y^  S3  =  Y^  S  =\^  Si,  the  altitude. 
This  distance  h  S^  laid  off  upon  a  perpendicular  drawn 
through  Y^  to  the  horizon  H  0  T  at  ?>,  gives  the  point  S4. 
203.  From  this  point  lines  drawn  to  the  several  van- 
ishing points  on  HOT  make  the  same  angles  one  with 
another  as  do  the  lines  0,  I,  0',  and  T,  and  since  I 
bisects  the  angle  between  O  and  0',  Y^'  is  easily  ascer- 
tained by  drawing  a  line  on  one  side  of  S4  Y^  at  the 
same  angle  that  S4Y^  already  makes  upon  the  other 
side  of  it. 


204.    In  these  figures  the  mirror  stands  at  an  angle 
with  the  plane  of  the  picture,  and  also  at  an  The  mirror 

.  vertical,  but 

angle   with  the  ground.     Ihe    case   is   some-  at  au  angle 

with  the  pic- 

what  simpler  when  the  latter  angle  is  90°,  the  ^^^^' 


136  MODERN   PERSPECTIVE. 

mirror  being  upright.  The  horizon  of  tlie  mirror  be- 
comes vertical,  and  the  normal  lines  are  horizontal.  If 
in  Figs.  39,  40,  and  41,  we  imagine  the  mirror  set  back 
into  a  vertical  position,  it  is  clear  that  V^  will  move  so 
as  to  coincide  with  V^,  S2  with  Sj,  andHLM'  with 
H  LZ ;  /  will  be  at  an  infinite  distance,  and  the  line  eg 
will  be  vertical.  But  the  esseutial  conditions  of  the 
problem  will  remain  unchanged,  and  the  horizon  of  the 
normal  plane,  the  point  where  the  given  line  pierces  the 
mirror,  and  the  vanishing  point  of  the  image  will  be 
obtained  as  above. 

205.  If  the  object  reflected  is  a  plane  surface,  its  re- 
The  refleo-     flcction  must  be  found  by  finding  the  images 

tion  of  a 

plane  figure,  of  the  liucs  that  bound  it,  and  their  vanishing 
points.  Tliese  vanishing  points  will  of  course  lie  in  a 
straight  line,  the  horizon  of  the  reflection  of  the  plane  ; 
this  horizon  can  be  drawn  as  soon  as  the  vanishing  points 
of  any  two  of  the  elements  of  the  plane  are  ascertained. 
It  is  convenient  to  take  one  of  these  elements  parallel  to 
the  mirror  (207). 

206.  To  obtain  the  reflection  of  a  point,  a  line  must 
The  reflec-  bc  passcd  tlirough  it,  and  its  image  obtained  as 
point.  above.  The  reflection  of  every  point  in  this 
auxiliary  line,  including  the  point  in  question,  is  then 
easily  found.  But  it  simplifies  the  problem,  as  we  shall 
presently  see,  to  take  the  auxiliary  line  either  parallel 
to  the  mirror  or  perpendicular  to  it  (209). 

207.  For  the  reflections  of  lines  which  are  parallel  to 


THE   PEUSPECTIVE    Ol<    REFLECTIONS.  137 

the  mirror,  being  in  every  part  as  far  behind  the  mirror 
as  the  lines  themselves  are  in  front  of  it,  are  Lines  parallel 
also  parallel  to  the  mirror  and  to  the  line  I,  in  mirror, 
which  the  normal  plane  in  which  they  lie  intersects  its 
surface.  They  are  accordingly  parallel  to  their  originals, 
and  have  the  same  vanishing  points. 

It  follows  that  if  a  plane  is  parallel  to  a  mirror,  all 
the  lines  in  it  retain  their  vanishing  points,  so  to  speak, 
in  the  reflection,  and  the  plane  retains  its  horizon,  i.  e., 
tlie  horizon  of  the  reflection  is  the  same  as  the  horizon 
of  the  plane  itself. 

208.  The  images  of  lines  perpendicular  to  the  mirror 
also    seem    to   retain   their   vanishing   point,  Lines  per- 

pendicular  to 

which  is  the  vanishing  point  of  normals,  ^^"^  ™"«'"- 
V^.  But  it  is  more  exact  to  say  that  tlie  other  vanish- 
ing point,  180°  away,  is  reflected  so  that  its  image  co- 
incides with  V^.  It  follows  that  the  reflections  of 
planes  perpendicular  to  a  mirror  have  the  same  horizons 
as  the  planes  themselves ;  for  the  vanishing  points  of 
the  parallel  elements  and  of  the  normal  elements  are 
alike  unchanged. 

209.  It  appears,  then,  that  the  reflections  of  lines  and 
planes  parallel  or  perpendicular  to  the  reflecting  sur- 
face have  the  same  vanishing  points  and  horizons  as  their 
originals,  whatever  the  position  of  the  reflecting  sur- 
face. 

210.  The  general  problem  of  the  reflections  of  lines 
and  planes,  whether  parallel,  perpendicular,  or  inclined 
to  the  mirror,  having  thus  been  discussed,  it  only  re- 


138  MODERN    PERSPECTIVE. 

mains  to  consider  the  special  cases  in  which  the  mirror 
itself  is  parallel  or  perpendicular  to  the  picture.  In 
both  these  cases  the  problem  is  a  very  simple  one. 

211.  When  the  mirror  is  perpendicular  to  the  2ola7ie  of 
The  mirror     thc  picture,  as  in  Figs.  42  and  43,  the  normals 

perpendicu-  .  .  •    i   • 

lartothe      arc    parallel   to    the    picture,  their  vamshm" 

plane  of  the  ^     ^^  ....  . 

picture.  point  is  at  an  infinite  distance,  and  the  horizon 
of  the  normal  plane  is  at  right  angles  to  that  of  the 
mirror.  The  horizon  of  the  plane  of  the  mirror  passes 
through  the  centre,  V^;  the  plane  drawn  tlirough  the 
station  point,  S,  intersecting  the  picture  in  this  horizon, 
is  perpendicular  to  the  picture,  and  lines  drawn  from  the 
station  point  to  the  vanishing  point  of  a  line  and  to  that 
of  its  reflection  strike  the  picture  at  equal  distances  from 
this  horizon. 

The  vanishing  point  of  the  image  of  a  line,  then, 
inclined  to  the  face  of  a  mirror  which  is  perpendicular 
to  the  picture,  is  as  far  on  one  side  of  the  horizon  of  the 
mirror  as  the  original  vanishing  point  is  on  the  other. 
on  a  line  at  right  angles  to  the  horizon. 

If  a  line  is  parallel  to  the  picture,  so  that  its  vanish- 
ing point  is  at  an  infinite  distance,  the  image  is  inclined 
to  the  horizon  of  the  plane  of  the  mirror  at  an  equal 
angle  on  the  other  side. 

212.  If  a  plane  is  inclined  to  a  mirror  that  is  per- 
pendicular to  the  picture,  one  element  of  the  plane  will 
nevertheless  lie  parallel  to  it,  and  that  element  will  retain 
its  vanishing  point,  whicli  will  lie  in  the  horizon  of  the 
plane  of  tlie  mirror  (209);  every  other  element  will, 
so  to  speak,  shift  its  vanishing  point  to  the  otlier  side 


THE  PERSPECTIVE  OF  REFLECTIONS.       1  o9 

of  the  horizon  of  the  rellecting  plane  (211).  The  horizon 
of  the  image  will  then  cross  the  horizon  of  the  mirror  at 
the  same  poi7it  with  that  of  its  original,  making  equal 
angles  on  the  other  side. 

213.  When  the  mirror  is  parallel  to  the  plane  of  the 
picture,  having  the  vanishing  point  of  its  axis  ^^^  ^^.^^^^ 
at  the  centre,  Y^,  a  line,  or  system  of  lines,  in-  Se  pJanfof 
clined  to  the  mirror  has  the  vanishing  point  "'*"  p^^^^^^^^. 
of  its  image  as  far  from  the  centrcV^^in  one  direction  as 
that  of  the  line  or  system  is  iii  the  other,  on  a  line  drawn 
through  the  centre.  If  a  line  goes  up  and  to  the  right, 
its  reflection  will  of  course  seem  to  go  down  and  to  the 
left  at  equal  angles. 

214.  It  follows  that  if  a  plane  is  inclined  to  a  mirror 
set  parallel  to  the  picture  the  horizon  of  its  image  is 
parallel  to  that  of  the  jjla^ie  itself  on  the  opposite  side  of 
the  centre,  V^,  and  equidistant  from  the  centre. 

215.  These  points  are  illustrated  in  Fig.  42.  The 
spokes  of  the  spinning  wheel  are  parallel  to 

the  riglit-hand  mirror,  the  axle  is  perpendicu- 
lar to  it.     All  retain  their  vanishing  points. 

The  box  on  the  left,  with  its  cover,  presents  four  sys- 
tems of  lines,  two  horizontal,  L  and  R ;  two  inclined,  N 
and  O.  The  reflection  in  the  floor  retains  V^  and  V^, 
but  exchanges  V^  and  V^  for  V^^  and  V^^  on  the  oppo- 
site side  of  the  Horizon  (209,  211). 

The  reflection  of  the  box  in  the  second  mirror,  on  the 
left,  has  for  vanishing  points  V^^,  V^^,  V^^,  V^^  across 


140  MODERN    PERSPECTIVE. 

the  horizon  of  the  mirror  H  X  A,  tlie  pLiue  LZg  haviug 
H  L  Z2  for  its  trace,  inclined  to  H  X  A  cc^ually  with 
HLZ(212). 

The  third  mirror,  parallel  with  the  picture,  in  like 
manner  oives  V^l  V^l  V^^  and  V^^  for  the  vanish- 
ing  points  of  the  main  lines  of  the  reflection  of  box 
(213). 

216.  The  reflections  are  themselves  reflected  just  like 
their  originals,  as  in  mirror  No.  2.  Y^^-^  is  the  vanish- 
ing point  of  the  lid  of  the  box,  reflected  first  in  the 
polished  floor  and  then  in  the  inclined  mirror. 

217.  Fig.  43  illustrates  these  principles  by  the  phe- 
Fig.  43.  Re-  nomena  of  reflections  in  water,  a  mirror  per- 

flectioas  in  ■*■ 

water.  pcndicular  to  the  j)lane  of  the  picture. 

The  steps  have  their  vanishing  point  at  V^  and  V^', 
their  reflections  at  V^'  and  V^,  respectively. 

218.  It  is  to  be  observed  that  the  phenomena  of  re- 
flection enable  us  to  determine  the  real  distance  of  iso- 
lated objects,  such  as  birds  or  distant  mountains,  the 
point  on  the  plane  of  the  water  directly  below  them 
being  midway  between  the  object  and  the  image.  The 
distance  of  this  point  is  the  horizontal  distance  of  the 
object. 

It  is  also  to  be  noticed  that  the  different  size  and  in- 
clination of  the  sticks  in  the  foreground,  which  are  not 
very  obvious  in  themselves,  are  made  conspicuous  by 
the  difference  of  their  reflections,  as  it  w^ould  be  indeed 
on  solid  ground  by  the  difference  of  their  shadows. 
These  objects  exhibit  very  clearly  the  relation  of  a  line 


THE   PERSPECTIVE   OF   REFLECTIONS.  141 

and  its  reflection  to  the  normal  plane  in  which  they 
lie,  and  to  the  line  in  which  this  plane  intersects  the 
mirror. 

That  the  horizontal  line  of  birds  should  be  reflected 
in  an  inclined  line,  and  the  inclined  line  of  birds  in  a 
horizontal  line,  is  easily  understood  by  observing  the 
line  of  dots  on  the  surface  of  the  water,  midway  between 
the  birds  and  their  reflections. 

Each  bird,  as  well  as  each  mountain  top,  and  indeed 
every  other  point,  has  its  reflection  as  far  below  the  sur- 
face of  the  water  directly  beneath  it  as  the  point  itself 
is  above.  The  plane  of  the  surface  of  the  water  is  sup- 
posed to  extend,  of  course,  beneath  the  land. 


CHAPTEE  XL 

THE  PERSPECTIVE  OF  CIRCLES. 

IN"  the  ten  preceding  chapters  we  have  considered  all 
the  principal  problems  of  plane  perspective  em- 
braced in  our  scheme.  That  is  to  say,  we  have  shown 
how  to  obtain,  upon  a  plane  surface,  the  perspective  rep- 
resentation of  a  straight  line  ;  whatever  the  position  of 
the  surface,  and  whatever  the  position  of  the  spectator, 
we  have  shown  liow  to  obtain  the  position,  magnitude, 
and  direction  of  the  representation  of  a  line,  when  the 
position,  magnitude,  and  direction  of  the  line  itself  are 
known.  The  problems  of  shadows  and  of  reflections 
have  also  been  fully  discussed,  so  far  as  concerns  plane 
surfaces..  Throughout  the  whole  investigation  it  has 
been  shown  that  the  problem  of  the  line  includes  the 
problems  of  plane  and  solid  figures  and  of  the  point. 
In  every  case  the  vanisliing  point  of  every  line  and  the 
vanishing  line,  or  horizon,  of  every  plane  has  been 
ascertained,  the  solution  being  considered  incomplete 
until  this  was  accomplislied. 

219.  The  only  lines  included  in  this  survey  have  ac- 
cordingly been  right  lines,  and  the  only  plane  or  solid 
figures  have  been  such  as  are  bounded  by  right  lines. 
Any  other  line  or  outline  in  perspective,  as  elsewhere  in 
geometry,  must  in  general  be  treated  as  a  series  of  points, 


THE    PERSPECTIVE   OF    CIRCLES.  143 

the  perspective  representation  of  each  point  being  ob- 
tained separately.  But  to  this  rule  the  circle,  here 
as  elsewhere,  constitutes  an  exception,  its  exceptional 
importance  making  it  worth  while  to  give  it  special 
consideration,  while  its  peculiar  geometrical  properties 
render  the  investigation  exceptionally  simple  and  easy. 

We  shall  find  also  that  the  study  of  the  circle  in 
perspective,  and  of  its  derivatives,  the  cylinder  and  the 
sphere,  introduces  a  new  set  of  most  interesting  phe- 
nomena, the  investigation  of  which  will,  in  the  two 
subsequent  chapters,  lead  to  theoretical  and  practical 
conclusions  of  the  first  importance. 

220.  The  perspective  representation  of  a  circle  will  ob- 
viously be  the  line  in  which  the  plane  of  the  „, 

'^  ^  The  perspec- 

picture  intersects  a  cone  of  rays  of  whicli  cird^^aconic 
the  vertex  is  in  the  eye  of  the  spectator,  at  ^^'^*^°°- 
the  station  point,  and  the  base  is  the  circle  itself.  The 
theory  of  conic  sections  establishes  the  feet  that  this 
line  of  intersection  will  be  a  circle,  ellipse,  parabola,  or 
hyperbola,  according  to  the  angle  at  which  the  plane  of 
the  picture  cuts  the  cone  of  the  rays,  and  this  whether 
the  axis  of  the  cone  be  at  ri^ht  anoles  to  the  circle  or 
inclined  to  it.  In  other  words,  the  cross-section  of  the 
cone,  perpendicular  to  its  axis,  may  be  either  a  circle  or 
an  ellipse.  If  the  secant  plane  is  parallel  to  the  base,  or 
equally  inclined  to  the  axis  in  a  contrary  direction, 
making  what  is  called  a  suh-contrary  section,  the  per- 
spective will  be  similar  in  shape,  though  not  of  course  in 
size,  to  its  original. 


144  MODERN   PERSPECTIVE. 

221.  This  is  illustrated  in  Fig.  44,  Plate  X.,  and  also 

by  the  three  figures  54,  55,  and  5G,  in  Plates 

Fig.  44. 

XL,  XII.,  and  XIIL,  all  of  which  show  how 
horizontal  circles,  whether  below  or  above  the  eye,  will 
appear  when  viewed  from  different  positions. 

In  Fig.  44  the  spectator  is  shown  as  regarding  the  cir- 
cular room  on  the  right  from  three  different  positions.  At 
S^  he  is  outside  the  room ;  the  plane  of  the  picture,  p  a, 
cuts  completely  across  the  cones  of  rays,  and  the  sections 
are  ellipses,  as  shown  at  A,  below.  At  S^  he  is  just  upon 
the  edge  of  tlie  circle ;  the  plane  of  the  picture,  p  h,  cuts 
the  cones  in  a  direction  parallel  to  one  side,  and  the  sec- 
tions are  parabolas,  as  seen  below  at  B.  At  S^  the  spec- 
tator is  fairly  within  the  circle,  and  the  intersection  of 
the  vertical  plane,  pc,  with  the  cones  of  rays  gives 
Figs  54  hyperbolas,  as  at  C.  Figs.  54,  55,  and  56, 
^'  ^^'  which  are  reproductions  of  engravings  of  cir- 
cular halls  in  the  Vatican  Palace,  excellently  illus- 
trate these  elliptical,  parabolic,  and  hyperbolic  lines : 
the  first  being  drawn,  presumably  with  a  camera  lucida, 
from  a  point  outside  the  circle ;  the  second,  from  a  point 
just  on  the  edge  of  the  room  ;  the  third,  from  a  point 
within  it. 

222.  Ficj.  44  also  illustrates  the  case  in  which  the 
cone  of  rays  is  intersected  by  the  plane  of  the  picture  in 
such  a  way  as  to  give  .a  sub-contrary  section.  The  small 
horizontal  circle  forming  the  eye  of  the  dome  is  the  base 
of  a  cone  of  rays  which  is  cut  by  the  plane  ap,  at  an 
angle  with  the  axis  of  the  cone  equal  to  that  made  by 
the  circle  itself,  but  taken  in  a  contrary  direction.     Both 


THE    PEllSPECTIVE    OF    CIRCLES.  145 

are  obviously  angles  of  45°  with  the  axis,  which  is 
accordingly  at  45°  with  the  horizon.  The  perspective 
of  the  circle  is  accordingly  a  circle  (220),  as  is  shown 
below.  In  Fig.  54,  also,  Plate  XI.,  the  perspective  of  the 
circle  at  the  top  of  the  dome  is  almost  exactly  circular. 

223.  Fig.  45  illustrates  more  fully  the  sub-contrary 
section  spoken  of  in  the  previous  paragraph, 

and  shown  in  Fig.  44.  Ai  Aj  shows  the  circle 
in  its  own  plane,  with  its  centre  beyond  the  axis  of  the 
cone  ;  Bi  B^  shows  its  perspective  in  the  plane  of  the  pic- 
ture, 20 p,  with  the  centre  of  the  cone  of  rays  below  its 
centre,  and  the  point  a,  representing  the  centre  of  the 
original  circle,  lower  still ;  Ej  Ei  shows  the  real  shape  of 
the  cross-section  EE,  taken  at  right  angles  with  the 
axis  of  the  cone.  This  is  an  ellipse,  whose  centre  coin- 
cides with  the  axis  of  the  cone  of  visual  rays,  the  centres 
of  both  circles  being  also  given,  one  on  one  side  and  one 
on  the  other.  The  projections  of  the  respective  centres 
are  shown  in  each  case,  at  a,  h,  and  e. 

The  line  D  D,  parallel  to  A  A,  shows  that  a  horizontal 
section  of  the  cone  taken  at  this  place  must  be  a  circle 
like  Aj  Ai ;  and  since  the  sub-contrary  section  at  B  B  is 
symmetrical  with  it,  about  the  axis  of  the  cone,  it  fol- 
lows that  Bi  Bi  also  must  be  a  circle. 

224.  It  is  to  be  noticed  that  the  ellipse  Ej  Ej  is  the 
appearance  that  the  circle  would  present'  from  the  sta- 
tion point,  S.  It  would  not  appear  as  a  circle,  though 
its  perspective  is  a  circle.  But  neither  does  this 
perspective  circle  appear  as  a  circle.     It,  too,  is  fore- 

10 


146  MODERN   PERSPECTIVE. 

shortened  into  an  ellipse  in  the  sight  of  a  spectator 
at  S  (230). 

225.  In  fact,  unless  a  circle  is  situated  just  at  the 
centre  of  the  picture,  the  ellipse  which  represents  it  in 
perspective  is  of  a  different  shape  from  the  ellipse  which 
it  presents  to  the  eye.  Horizontal  circles,  for  instance, 
always  present  to  the  eye  horizontal  ellipses  ;  ellipses, 
that  is  to  say,  whose  major  axes  are  horizontal.  But  in 
perspective  such  circles,  unless  just  above  or  below  the 

centre,  V^,  have  their  axes  inclined,  as  we  may 

Fig.  47. 

see  in  Fig.  47.  Yet  these  oblique  ellipses, 
when  seen  from  the  proper  position,  the  station  point 
in  front  of  V^,  are  themselves  apparently  changed  in 
shape  by  the  effect  of  j)erspective,  and  foreshortened  into 
horizontal  ellipses. 

This  seeming  distortion  in  the  perspective,  which 
makes  the  outline  in  tlie  drawing  of  a  different  shape 
from  the  apparent  outline  of  the  thing  drawn,  will  form 
the  subject  of  the  next  chapter. 

226.  In  general,  of  course,  the  station  point  is  out- 
side the  circle  to  be  represented,  so  that  practically  the 
problem  of  putting  a  circle  into  perspective  is  this  :  to 
find  the  ellipse  which  represents  it. 

The  simplest  and  generally  the  most  efficient  way  to 
To  draw  the    do  tliis  is  to  supposc  a  square  or  octagon  to  be 

perspective  ,  m       i       i  i  •  •      i 

ellipse.  circumscribed  about  the  given  circle,  at  any 

angle  that  may  be  most  convenient.  The  centres  of 
these  sides  give,  of  course,  four  or  eight  points  of  the 
required  ellipse.     As  the  direction  of  the  sides  gives  the 


THE   PERSPECTIVE   OF   CIRCLES.  147 

direction  of  the  ellipse  at  these  points,  it  can  easily  be 
drawn  with  sufficient  accuracy  for  practical  purposes. 
If  greater  accuracy  is  required  the  number  of  sides  of 
the  circumscribing  polygon  can  be  increased. 

This  is  illustrated  in  Fig.  47.  Fig-  47. 

227.  But  in  order  to  draw  an  ellipse  with  absolute  pre- 
cision it  is  necessary  to  find  its  centre,  and  the  direction 
and  length  of  its  axes  or  principal  diameters. 

It  is  obvious  from  Fig.  44  and  from  Fig.  54,  Plate  XT., 
that  the  perspective  of  the  centre  of  a  circle  its  centre 

and  extreme 

does  not  coincide  witli  the  middle  point  of  pomts. 
the  ellipse,  as  indeed  it  cannot,  since  the  fixrther  half  of 
a  circle  must  appear  smaller  than  the  nearer  half,  and 
its  radii  shorter.  Neither  do  the  extreme  points  of  the 
ellipse  represent  extreme  points  of  the  circle ;  the  high- 
est point  in  the  perspective  of  an  arch,  as  may  be 
seen  in  the  arched  windows  and  niches  of  Fig.  54, 
is  the  perspective,  not  of  the  highest  point  in  tlie  arch, 
but  of  a  lower  point  nearer  tlie  spectator.  In  fact, 
although  a  circle  put  into  perspective  appears  as  an 
ellipse,  the  diameters  of  the  circle  do  not  become  diam- 
eters of  the  ellipse,  but  chords,  wliich  intersect  at  a 
point  situated  beyond  the  centre.  On  the  other  hand, 
the  diameters  of  the  ellipse,  meeting  and  intersecting  at 
its  centre,  are  the  perspectives  of  chords  of  the  circle, 
which  meet  and  intersect  at  a  point  within  the  circle 
nearer  to  the  spectator  than  its  centre.  The  tangents 
at  the  extremities  of  each  diameter  of  the  circle  are 
parallel;  but  their  perspectives  of  course  converge  to 
a  vanishing  point ;  and  since,  for  each  circle^  all  these 


148  MODERN   PERSPECTIVE. 

tangents  lie  in  the  same  plane,  these  vanishing  points  all 
lie  in  the   same   strai^jht   line,  which   is  the 

Fig.  46. 

horizon  of  that  plane.     See  Fig.  46. 

228.  These  phenomena  afford  a  curious  illustration  of 

certain  well  known  geometrical  properties  of 
and  Polar      the  cllipsc.     If  a  point  be  taken  at  random 

anywhere  within  a  circle  or  an  ellipse,  and 
chords  be  drawn  through  that  point,  then  the  tangents 
drawn  from  the  extremities  of  each  chord  will  have 
their  point  of  intersection  upon  a  right  line.  This  line 
is  called  a  polar  line,  the  point  assumed  being  called  a 
pole.  As  a  mere  geometrical  proposition  this  seems  to 
have  no  special  significance  ;  but  the  phenomena  of  per- 
spective give  it  meaning.  For  a  circle  seen  in  perspec- 
tive becomes  an  ellipse,  its  centre  a  pole,  its  diameters 
chords,  and  the  polar  line  is  tlie  horizon  upon  which 
meet  the  tangents  drawn  from  the  extremities  of  its 
diameters. 

229.  Fig.  46  exhibits  these  relations,  and  illustrates 

also  the  further  proposition,  which,  however. 

To  find  the  ^ 

centre  of  a     docs  uot  sccm  to  admit  of  similar  interpretation, 

given  ellipse.  ^ 

that  if  lines  are  drawn  from  the  points  where 
the  tangents  meet  through  the  middle  of  the  chords 
they  will  pass  through  the  centre  of  the  ellipse.  This 
property  we  shall  find  a  use  for  presently  (235,  238). 

230.  Although  in  Fig.  44,  A,  and  in  Fig.  45,  the  small 
Fig.  44,  A      vertical  circle  is  the  perspective  of  the  horizon- 

'^*  tal  circle  at  the  top  of  the  dome,  so  that,  when 

seen  from  the  station  point,  the  two  circles  seem  to  co- 
incide, yet,  as  we  have  seen  (223),  their  centres  do  not 


THE  PERSPECTIVE  OF  CIRCLES.  149 

coincide,  and  neither  of  them  coincides  with  the  axis  of 
the  cone.  The  centre  of  each  circle  appears  as  a  pole  of 
the  other.  If  the  cone  be  cut  by  a  plane  at  right  angles 
with  its  axis,  as  at  E  E,  Fig.  45,  the  section  will  be  an 
ellipse,  El  E^,  of  which  the  axis  of  the  cone  will  give  the 
centre,  and  of  which  the  centre  of  the  upper  circle  will 
be  a  pole  situated  just  below  the  axis,  and  the  centre  of 
the  lower  circle  will  be  another  pole  just  above  it.  In 
the  horizontal  circle  the  centres  of  the  ellipse  and  of  the 
vertical  circle  become  poles,  and  in  the  vertical  circle 
the  centres  of  the  ellipse  and  of  the  horizontal  circle 
become  poles. 

An  ellipse,  of  course,  may  by  perspective  be  fore- 
shortened into  a  circle,  —  the  centre  of  the  ellipse  be- 
coming a  pole,  just  as  a  circle  appears  like  an  ellipse. 

231.  It  will  be  noticed  in  figures  54,  55,  and  56, 
and  also  in  Fig.  47,  that  the  vertical  circles  whose 
centres  are  on  the  Horizon,  and  the  horizontal  circles 
whose  centres  are  exactly  above  or  below  the  centre  of 
tlie  picture, V^, lie  symmetrically  on  the  paper;  that  is 
to  say,  that  their  principal  diameters,  their  major  and 
minor  axes,  are  vertical  and  horizontal,  but  that  other 
ellipses  have  their  axes  more  or  less  inclined. 

Another  and  more  comprehensive  statement  of  this 
phenomenon  is  this  :  that  if  a  line  drawn  through  the 
centre  of  a  circle  normal  to  its  plane,  like  the  axle  of  a 
wheel,  crosses  the  centre  of  the  picture,  Y^,  one  axis  of 
the  ellipse  that  represents  the  circle  will  coincide  with 
this  line,  and  will  also  be  directed  towards  Y^,  and  tho 


150  MODERN   PERSPECTIVE. 

other  will  be  at  right  angles  to  it.  Other  circles,  which 
do  not,  as  it  were,  thus  face,  the  centre,  V*^,  will  be  pro- 
jected in  ellipses  the  direction  of  whose  axes  it  is  more 
difficult  to  determine. 

Fig.  47  shows  a  number  of  circles,  three  of  which, 
A  A,  B  B,  and  D  D,  are  vertical,  and  accordingly  appear 
in  plan  as  right  lines,  and  two,  E  E  and  F  F,  are  hori- 
zontal. These  appear  in  perspective  at  Aj  Ai,  Bi  B^, 
Di  Di,  El  El,  and  Fi  Fi,  respectively.  In  all  of  these 
except  the  last,  one  of  the  principal  axes  passes  through 
the  centre,  V^.  In  Fi  Fi,  and  also  in  the  circles  A2  A2  and 
B2  B2,  the  position  of  the  principal  axes  is,  so  to  speak, 
accidental. 

232.  Let  us  first  take  the  case  of  the  ellipses  which 
Ellipses  that  I'^preseut  circles  tiiat  do  face  the  centre,  V^,  and 
JeSre.*         which  accordingly  lie  symmetrically  about  a 

Fig.  47.  normal  line  joining  this  point  with  the  per- 
spective of  their  centres.  Tliis  line  will,  of  course,  also 
pass  through  the  centre  of  the  ellipse,  as  in  Ai  Ai,  Bi  Bj, 
and  El  Ei,  Fig.  47. 

233.  If,  as  in  Ai  Ai  and  Bi  Bi,  two  of  the  sides  of  the 
circumscribing  square  or  octagon  are  parallel  to  the 
plane  of  the  picture,  and  their  perspectives  consequently 
parallel  to  each  other  and  perpendicular  to  the  normal 
line,  the  line  joining  their  middle  points  will  be  the 
minor  axis  of  the  ellipse  ;  the  major  axis  will  cross  it  at 
its  middle  ])oint,  and  it  will  only  remain  to  ascertain  the 
length  of  this  major  axis. 

234.  Fig.  47  shows  how  this  is  done.     Let  A  A,  B  B, 


THE   PERSPECTIVE   OF   CIRCLES.  151 

and  D  D,  in  the  plan,  be  three  parallel  and  similar  circles, 
all  touching  the  plane  of  the  picture,  and  the  last,  as 
appears  from  the  perspective  below,  standing  edgewise 
to  the  spectator  at  S.  If  the  space  betw^een  them  were 
filled  up  with  other  such  circles,  they  would  all  together 
constitute  an  elliptical  cylinder,  the  apparent  vertical 
dimension  of  which  would  be  the  apparent  height  of 
each  of  the  circles  and  of  the  major  axis  of  the  ellipses 
that  represent  them.  Now  let  the  plane  containing  the 
circle  D  D  and  the  station  point,  S,  be  revolved  into  the 
plane  of  the  picture  around  the  vertical  line  H  li  Z,  in 
which  the  two  planes  intersect.  S  will,  of  course,  fall  at 
the  point  of  distance  D^ ;  the  circle  D  D  will  appear  of  its 
true  shape  and  size  ;  lines  drawn  from  D^  tangent  to  the 
circle  D  D  thus  revolved  will  determine  the  highest  and 
lowest  points  visible  from  S,  and  the  points  where  these 
lines  cut  the  line  HE  Z  will  show  the  perspective  of  these 
points  in  the  plane  of  the  picture,  and  fix  the  apparent 
height  of  D  D.  The  circles  B  B  and  A  A  will  appear  to  be 
of  the  same  height  as  D  D,  and  D^  D^  will  be  the  length 
of  the  major  axes  of  the  ellipses  that  represent  them. 

235.  When,  as  in  the  case  of  the  circle  E  E,  the  tan- 
gent lines  are  not  parallel  to  the  picture,  the  square  or 
octagon  that  encloses  the  circle  being  in  angular  per- 
spective, instead  of  being  in  parallel  perspective,  as  in 
the  previous  instance,  the  centre  of  the  ellipse  must  be 
obtained  as  above  explained  (229)  and  shown  in  Fig.  46, 
by  bisecting  two  of  the  chords,  and  drawing  lines  from  the 
vanishing  points  of  their  tangents.  The  principal  axes  of 
the  ellipse  may  then  be  drawn,  one  towards  the  centre,  V^ 


152  MODERN   PERSPECTIVE. 

and  the  other  parallel  to  the  picture,  that  is  to  say, 
parallel  to  the  horizon  of  the  plane  in  which  the  circle 
lies  (38 ).  Tlie  length  eeof  the  latter,  or  major  axis,  may 
then  be  found  by  direct  projection,  as  in  the  figure. 

236.  To  find  the  length  of  the  minor  axis,  the  major 
axis  and  one  point  of  the  ellipse  being  given,  it  is  only 

necessary  to  employ  the  usual  device,  shown 

Fig.  48. 

in  Fig.  48,  founded  upon  the  proposition  that 
if  a  semicircle  be  erected  on  the  major  axis  of  an  ellipse 
the  distance  of  the  different  points  of  the  ellipse  from 
this  axis  will  be  proportional  to  that  of  the  correspond- 
ing points  of  the  semicircle.  Thus,  in  the  figure,  one 
point  X  being  given  on  the  ellipse,  and  x'  and  b'  obtained 
on  the  circle,  the  point  h  at  the  extremity  of  the  minor 
axis  is  easily  found,  since  the  chords  h  x  and  h'  x'  meet  on 
the  line  of  the  major  axis  prolonged. 

237.  When  the  ellipse  does  not  lie  opposite  the 
Ellipses  that  ccutre,  V^,  as  is  the  case  with  Fi  Fj,  and  with 
the  centre.     Ag  A2  aud  B2  B2,  iu  Fig.  47,  thc  normals  drawn 

Fig.  47.  through  the  centres  of  these  circles  not  pass- 
ing across  the  centre  of  tlie  picture  (231),  the  only  way 
to  obtain  the  principal  diameters,  or  axes,  is  first  to  ob- 
tain a  pair  of  conjugate  diameters.  Conjugate  diameters 
are  diameters  each  of  which  is  parallel  to  the  tangents 
drawn  through  the  extremities  of  the  other.  The  axes 
are  that  pair  of  conjugate  diameters  which  are  at  right 
angles  with  each  other ;  and  one  is  always  the  longest 
diameter  that  can  be  drawn  in  a  given  ellipse,  and  the 
other  the  shortest. 


THE   PEKSPECTIVE    OF    CIRCLES.  153 

238.  The  quickest  way  to  obtain  a  pair  of  conjugate 
diameters  in  oblique  ellipses  such  as  A2  A2,  B2  B2,  and 
Fi  Fi  is  to  construct  first  such  horizontal  and  vertical 
ellipses  as  Aj  Aj,  Bi  B^,  and  Ei  Ej,  respectively,  opposite 
the  centre,  and  to  obtain  their  principal  axes  as  just  de- 
scribed. If  now  the  centre  of  each  ol)lique  ellipse  is 
found,  as  above  (229),  and  lines  passed  through  it  per- 
spectively  equal  and  parallel  to  these  principal  diameters, 
they  will  be  conjugate  diameters  of  the  oblique  ellipses. 
They  will  not  be  at  right  angles,  but  each  will  be  parallel 
to  the  tangents  drawn  at  the  extremities  of  the  other; 
one  will  be  parallel  to  the  plane  of  the  picture  and  tlie 
other  perpendicular  to  it,  and  directed  to  the  centre,  V^. 

239.  Conjugate  diameters  of  the  ellipses  A2  A2,  B2  B2, 
and  Fi  Fj  being  thus  determined,  their  principal  diame- 
ters or  axes  can  now  easily  be  obtained  by  the  well 
known  method  of  shadows,  as  in  the  figure. 

240.  Fig.  49,  a,  shows  this  ingenious  device  more  in 
detail.    It  is  called  the  Method  of  Shadows,  be- 

Fig.  49,  a. 

cause  the  ellipse  is  regarded  as  the  shadow  or 
projection  of  a  circle.     The  process  is  this  :  — 

A  tangent  being  drawn  at  the  extremity  of  one  diam- 
eter parallel  to  its  conjugate,  a  circle  is  erected  .  ^^^ 

also  tangent  at  the  same  point,  of  such  size  J"g7ifj[n 

that  the   ellipse  might  be  its   shadow.     The  f^Zn^r 

shadow  of  every  diameter  of  the  circle  will  be  Srsaregi^en 

•^  bv  the 

a  diameter  of  the  ellipse,  and  the  shadows  of  nlethoa  of 

^  shadows. 

any  two  diameters  of  the  circle  which  are  at 

right  angles  with  each  other  will  be  conjugate  diameters 


154  MODERN   PERSPECTIVE. 

of  the  ellipse,  since  the  tangents  at  the  extremity  of  one 
will  be  parallel  to  the  other.  The  given  conjugates  of 
the  ellipse  are  shadows  of  those  diameters  of  the  circle 
which  are  perpendicular  to  and  parallel  with  the  tan- 
gent line  common  to  both  circle  and  ellipse.  Since  the 
shadow  of  the  diameter  parallel  to  the  tangent  is  also 
parallel  to  the  tangent,  the  diameter  and  its  shadow  are 
parallel  to  each  other  and  must  be  of  the  same  length. 
This  fixes  the  size  of  the  circle,  the  distance  of  whose 
centre  from  the  end  of  one  diameter  is  equal  to  half  the 
length  of  its  conjugate. 

241.  It  now  only  remains  to  find  in  this  circle  a  pair 
of  diameters  at  rio-ht  andes  to  each  other  whose  shadows 
will  also  be  at  right  angles.  But  since  it  is  plain  that  if 
these  diameters  are  prolonged  till  they  reach  the  tangent 
line  their  shadows  will  also  be  prolonged,  and  will  reach 
the  tangent  line  at  the  same  points,  the  problem  becomes 
a  very  simple  one.  It  is  only  necessary  to  find  two  points 
on  the  tangent  line  which  make  right-angled  triangles  both 
with  the  centre  of  the  circle  and  with  the  centre  of  the 
ellipse ;  that  is  to  say,  two  points  such  tliat  tlie  portion 
of  the  tangent  lying  between  them  shall  be  the  common 
diameter  of  two  semicircles  passing  respectively  through 
these  two  centres.  The  common  centre  of  these  semi- 
circles must  be  a  point  on  the  tangent  line  equidistant 
from  the  two  centres  ;  a  point  easily  found  by  erecting 
a  perpendicular  upon  tlie  middle  of  the  line  connecting 
them,  as  is  done  in  the  figure.  Semicircles  struck  from 
this  point  c  as  a  centre,  with  a  radius  equal  to  its  dis- 
tance from  the  centre  of  the  circle  or  of  the  ellipse,  give 


THE   PERSPECTIVE   OF   CIRCLES.  155 

the  points  a  and  h,  through  which  diameters  can  be 
drawn  in  the  circle  whose  shadows,  drawn  through  the 
same  points  to  the  centre  of  the  ellipse,  are  axes  or 
principal  diameters  of  the  ellipse,  both  sets  of  diameters 
making  right  angles  with  each  other. 

242.  As  the  centre  of  the  ellipse  is  the  shadow  of  the 
centre  of  the  circle,  the  line  that  joins  these  centres  may 
be  considered  to  give  the  direction  of  the  light,  and  lines 
drawn  parallel  to  it  through  the  extremities  of  the  diam- 
eters of  the  circle  will  give  the  extremities  of  the  corre- 
sponding diameters  of  the  ellipse,  or  the  length  of  the 
axes. 

243.  This  operation,  though  long  in  the  description, 
is  simple  in  practice,  and  requires  very  few 
constructive  lines,  as  is  seen  in  Fig.  49,  h,  in 

which  the  operation  just  described  is  repeated  with  no 
more  construction  lines  than  are  necessary. 

This  method  is  used,  as  has  been  said,  in  finding  the 
axes  of  A2  A2,  B2  B2,  and  Fj  Fj,  and  all  the  necessary  con- 
struction lines  are  given  in  the  figure. 

244.  When  only  a  portion  of  a  circle  is  to  be  put  into 
perspective  it  is  generally  best  to  construct  the  ^^^  perspec- 
ellipse  which  represents  the  whole  circle,  and  circies,^and° 
then  to  use  so  much  of  it  as  may  be  required. 

This  is  eminently  the  case  in  sketching  from  nature  or 
from  the  imagination,  where  it  is  difficult  to  deter- 
mine the  character  of  the  perspective  curve  without  aid 
from  geometrical  considerations.  In  drawing  pointed 
arches,  for  instance,  the  character  of  the  intersecting  arcs 


156  MODERN  PERSPECTIVE. 

is  best  ascertained  by  completing  the  circles  of  which 

they  form  a  part,  as  in  Fig.  50.     An  inspection  of  the 

fifijure  shows  that  when  the  arch  is  above  the 

Fig.  50. 

eye  the  nearer  half  is  represented  by  the  part 
of  an  ellipse  at  which  the  curvature  is  the  most  rapid, 
near  the  extremity  of  the  major  axis ;  and  the  further 
half  by  the  flattest  portion,  near  the  extremity  of  the 
minor  axis.  When  the  circle  is  below  the  eye  the  nearer 
part  is  the  flattest. 

This  figure  also  shows  that  when  a  row  of  circles  is 
put  into  perspective  their  major  axes  are  not  parallel, 
their  inclination  to  the  horizon  of  the  plane  in  which  the 
circles  lie  diminishing  as  they  approach  it.  This  is 
illustrated  also  in  Fig.  53. 

245.  Figs.  51,  52,  and  53  show  three  different  ways 
Concentric  ^^  drawing  conccutric  circles.  Since  concen- 
circies.  ^j.|^  circles  have  the  same  centre,  the  ellipses 
which  constitute  their  perspectives  have  of  course  the 
same  pole  and  polar  line ;  but  the  ellipses  have  not  the 
same  centre,  nor  are  their  axes  parallel. 

246.  The  first  metliod  is  shown  in  Fig.  51 ;  it  is  ap- 

plicable to  the  case  where  the  perspective  of  a 

circle  is  obtained  by  means  of  a  circumscribed 

square  or  polygon.     A  second  circle,  concentric  with  the 

first,  is  easily  obtained  by  means  of  a  concentric  polygon, 

as  shown. 

247.  Fig.  52  shows  how  the  second  ellipse  can  be 

found  when  the  first  lias  been  alreadv  deter- 

Fig.  52. 

mined  in  any  way.     Let  a  line  of  measuies  be 


THE   PERSPECTIVE   OF   CIKCLES.  157 

drawn  througli  the  pole  which  is  the  perspective  of  the 
centre  of  the  circle,  parallel  to  the  polar  line,  or  horizon  of 
the  plane  in  which  the  circle  lies.  If  now  any  chord  a  a, 
representing  a  diameter  of  the  circle,  be  drawn  through 
this  pole,  and  lines  be  drawn  from  its  extremities  to  any 
point  V  upon  this  line  or  horizon,  it  will  cut  the  line  of 
measures  at  two  points,  a'  and  a',  wliose  distance  from 
the  pole  is  the  same.  If  now  two  other  points,  h'  and  h', 
be  taken,  also  equidistant  from  the  pole,  and  lines  be 
drawn  through  them,  the  points  1)  and  h,  in  which  they 
intercept  the  same  chord,  will  be  points  of  an  ellipse 
which  represents  a  circle  concentric  with  the  given  cir- 
cle, and  as  much  smaller  as  V  h'  is  smaller  than  a!  a. 
In  the  ficjure  the  radius  of  the  smaller  circle  is  one  half 
the  radius  of  the  larger  one,  V  h'  being  one  half  of  a  o!. 
It  is  obvious  that  since  the  lines  meeting  at  V  are  par- 
allel in  space,  the  lines  a  a  and  a!  a!  are  divided  propor- 
tionally. Any  number  of  points  can  be  obtained  in  the 
same  way  as  h  h. 

248.  A  third  method  of  putting  concentric   circles 
into  perspective  is  shown  in  Fii^^.  53,  —  a  fic^- 

ure  which,  like  Fig.  47  in  plate  X,  shows  three 
equal  circles,  A  A,  B  B,  and  D  D,  lying  in  parallel  planes 
and  equally  distant  from  the  picture,  the  last  of  which 
stands  cdgevme  to  the  spectator,  so  that  it  coincides  with 
a  portion  of  H  E  Z,  the  horizon  of  the  parallel  planes.  In 
the  figure  it  is  supposed  that  A  A  is  the  given  circle, 
concentric  with  which  it  is  required  to  draw  another 
circle  E  E. 

249.  To  effect  this  the  circle  D  D  is  first  found  by 


158  MODERN    PERSPECTIVE. 

cutting  off  from  H  E  Z  a  portion  equal  in  height  to  A  A, 
this  height  being  measured  above  and  below  a  line  pass- 
ing through  the  centres  of  the  three  circles.  Parallel 
to  this  line  let  a  second  line  be  drawn  through  any 
point,  1,  of  the  circle  A  A,  to  the  corresponding  point, 
3,  of  the  circle  D  D ;  and  upon  this  line  let  any  con- 
venient point,  as  2,  between  1  and  3,  be  taken  as  the 
corresponding  point  of  a  third  circle,  B  B.  If  now  a 
fourth  point,  4,  be  taken  upon  the  line  through  the 
centres,  the  line  4  2  5  will  be  an  element  of  a  cone 
whose  vertex  is  at  4  and  whose  base  in  the  plane  of  the 
circle  A  A  is  a  circle  concentric  with  that  circle.  The 
intersection  of  the  line  4  2,  prolonged,  with  a  radius  of 
A  A  drawn  through  the  point  1,  fixes  the  point  5,  in  the 
circumference  of  the  circle  E  E.  By  drawing  other  lines, 
parallel  to  the  line  12  3,  through  other  points  of  A  A, 
any  number  of  other  points  in  B  B  and  E  E  may  now 
easily  be  obtained. 

250.  The  radii  of  the  circles  E  E  and  A  A  (or  B  B) 
are  obviously  proportional  to  the  distance  of  the  point  4 
from  the  centres  of  E  E  and  B  B,  and  also  to  the  chords 
drawn  through  tlie  common  centre  of  A  A  and  E  E  par- 
allel to  D  D,  that  is,  to  the  trace  T  E  Z.  By  changing 
the  position  of  4,  the  size  of  E  E  may  be  made  larger  or 
smaller. 

251.  It  will  be  noticed,  1st,  that  this  method  not 
only  gives  the  means  of  finding  a  larger  circle  con- 
centric with  A  A,  and  lying  in  the  same  plane,  but 
also  of  finding  an  equal  circle,  B  B,  lying  in  a  paral- 
lel plane ;  2d,  that  it  makes  no  difference  in  what  direc- 


THE   PERSPECTIVE   OF   CIRCLES.  159 

tion  the  axis  of  the  cone  passing  through  the  three 
centres  is  oiiginally  drawn,  provided  it  is  parallel  to 
the  plane  of  the  picture ;  and  3d,  that  this  method  is 
as  serviceable  in  drawing  a  concentric  circle  smaller 
tlian  the  given  circle  as  in  drawing  a  larger  one ;  for 
if  E  E  were  the  given  circle  a  reversed  process  would 
give  A  A. 

252.  As  to  the  figures  54,  55,  and  56,  in  Plates  XL, 
XIL,  and  XIII.,  it  is  perhaps  worth  while  to  ^j  ^  ^^ 
say  that  the  excessive  distortion  apparent  in  ^'^^' 
them  is  due  simply  to  the  fact  that  the  station  point, 
or  proper  position  of  the  spectator,  is  in  each  of  them 
within  three  or  four  inches  of  the  page.  This  is  within 
the  limits  of  distinct  vision.  But  by  looking  through  a 
pin-hole  in  a  card  the  prints  can  be  distinctly  seen  when 
held  even  at  the  end  of  one's  nose ;  and  when  so  viewed 
it  will  be  seen  that  not  only  the  ellipses  in  Fig.  54,  but 
the  parabolas  and  hyperbolas  in  Figs.  55  and  56,  look 
like  circles,  as  they  should.  The  apparent  distortion 
entirely  disappears. 


CHAPTER  XII. 

DISTORTIONS   AND   CORRECTIONS. — THE   HUMAN  FIGURE. 

253.  It  has  been  pointed  out  in  the  previous  chapter 
that  the  perspectives  of  circles  often  look  very  queer ; 
the  ellipses  by  which  they  are  represented  seem  unac- 
countably and  even  unnaturally  inclined,  their  principal 
axes  slanting  in  directions  difficult  to  anticipate.  The 
effect  of  this  is  particularly  objectionable  when  the  circle 
forms  the  base  of  a  cylinder  or  when  it  is  horizontal. 
The  base  of  a  cylinder  always  presents  the  appearance 
of  an  ellipse  whose  major  axis  is  at  right  angles  with 
the  axis  of  the  cylinder,  and  it  is  offensive  to  find  it 
drawn  otherwise,  as  in  perspective  often  happens. 

254.  A  horizontal  circle  always  appears  to  the  eye  as 

a  horizontal  ellipse,  as  an  ellipse,  that  is  to 

Plate  XIV.  ...  11- 

say,  whose  major  axis  is  parallel  to  the  hori- 
zon and  whose  minor  axis  is  perpendicular  to  it,  and  it 
is  extremely  unpleasant  to  see  it  drawn  with  the  axes 
^.    ^Q  inclined.      This  is  illustrated  in  Plate  XIV., 

Fig.  58.  ' 

Horizontal      ^'^S-  ^^>  ^J  ^^^^  pcrspcctive  plan  of  the  capi- 
circies.  ^,^-^  ^^  ^^^g  ^^^^  ^£  ^l^g  figure  and  that  of  tlie  base 

at  the  bottom.     The  effect  of  this  would  be  so  disagree- 
able if  the  curves  of  the  capital  and  base  were  inclined  in 


DISTORTIONS   AND    CORRECTIONS.  161 

like  manner,  that  it  is  customary  to  introduce  a  certain 
correction,  as  it  is  called,  as  is  done  in  the  figure.  These 
lines  are  accordingly  drawn  as  horizontal  ellipses,  just  as 
if  these  objects  faced  the  centre  of  the  picture  (231). 

255.  Fig.  59,  c,  still  further  illustrates  this  point, 
showinoj  that  in  the  column  at  the  centre  of 

.  .  Fig.  59,  c. 

the   picture   the   ellipses  are  horizontal,  and 

^  ^  Distortions 

that  the  others  are  more  and  more  inclined  ejnnders 
as  they  are  farther  removed  from  it,  which  ^^^^  ^p^^'"^^- 
looks  like  an  unnatural  distortion.  In  this  figure,  more- 
over, the  outer  columns,  wliich  as  seen  from  the  station 
point  at  S  would  look  the  smallest,  since  they  are  far- 
thest from  the  eye,  are  on  the  contrary  drawn  larger  in 
diameter,  an  apparent  distortion  even  more  offensive 
than  the  other.  So  with  the  spheres  by  which  the 
columns  are  surmounted.  The  outline  of  a  sphere  al- 
ways looks  like  a  circle ;  it  is  not  agreeable  to  find  it 
drawn  as  an  ellipse.  But  in  perspective  it  must  always 
be  an  ellipse,  unless  its  centre  is  just  at  the  centre  of 
the  picture ;  for  the  perspective  representation  of  the 
sphere  is  the  section  of  a  right  cone  with  a  circular 
base,  the  base  of  the  cone  being  that  great  circle  of  the 
sphere  which  separates  tlie  side  of  the  sphere  one  can 
see  from  the  further  half  of  the  sphere  that  he  cannot 
see ;  it  must  always  be  an  ellipse  unless  the  axis  of  the 
cone  is  perpendicular  to  the  plane  of  the  picture. 

256.  Of  course  all  these  distortions  disappear  when 
the  eye  is  at  the  station  point,  at  a  proper  distance  in 
front  of  the  picture,  opposite  the  centre  V^.  From  that 
point  of  view  the  perspective  lines  exactly  cover  and 

11 


162  MODERN    PEKSPECTIVE. 

coincide  with  the  oulines  of  the  objects.  But  practi- 
cally it  is  impossible  for  the  spectator  always  to  be  ex- 
actly at  the  station  point,  and  since  from  every  other 
point,  circles,  cylinders,  and  spheres  appear,  in  general, 
to  be  more  or  less  distorted  in  the  manner  we  have  just 
seen,  it  is  customary  here  also  to  apply  corrections. 
These  corrections  are  palpable  violations  of  the  rules  of 
perspective  made  in  order  to  avoid  the  disagreeable  con- 
sequences of  deserting  the  station  point.    They 

Corrections.  .        .        ,  .  n   i        •  i      •      i  i 

consist  111  drawing  all  horizontal  circles  as  hor- 
izontal ellipses,  whether  opposite  the  centre  of  the  pic- 
ture or  not ;  in  always  drawing  the  elliptical  representa- 
tion of  the  base  of  a  cylinder  at  right  angles  with  the 
cylinder  itself ;  and  in  drawing  all  spheres  as  circles.  If 
a  row  of  columns,  moreover,  is  parallel  to  the  picture, 
they  are  always  made  of  the  same  diameter,  as  if  seen  in 
elevation,  and  if  the  direction  of  the  row  is  slightly  in- 
clined to  the  picture  care  is  taken  to  diminish  their 
width  a  little  as  they  recede. 

257.  Fig.  59,  a,  illustrates  these  corrections,  and  shows 

further  how  the  same  treatment  is  sometimes 
extended  to  the  octagon.  The  right-hand  side 
of  the  octagonal  figure  at  the  top  is  drawn  steeper  than 
it  ought  to  be,  not  being  directed  to  its  proper  vanishing 
point,  in  order  to  remedy  the  apparent  distortion  seen  in 
the  corresponding  figure  below. 

258.  Fig.  59,  a,  shows  also  the  effect  of  applying  to 
vertical  circles  and  semicircles  the  same  corrections  as 
to  horizontal  ones.  The  circular  window  which  in  the 
figure  below  is  drawn  in  true  perspective  as  an  oblique 


DISTORTIONS   AND    CORRECTIONS.  163 

ellipse  is  here  shown  as  a  vertical  one,  a  change  which 
will  probably  be  regarded  by  most  persons  as  an  im- 
provement. The  effect  of  a  similar  correction  in  the 
semicircular-window  head  beyond  is  less  happy;  it 
makes  the  nearer  half  look  much  too  big,  and  ob- 
viously throws  the  imposts,  or  points  where  the  arch 
begins,  quite  out  of  level. 

259.  Since  these  so-called  corrections  change  and 
generally  diminish  the  apparent  size  of  the  circles,  cyl- 
inders, and  spheres  to  which  they  are  applied,  the  rela- 
tion of  these  objects  to  other  objects  is  necessarily 
changed  at  the  same  time.  In  the  first  place,  more  of 
the  background  has  to  be  shown  than  can  really  be 
seen.  In  the  figure,  for  example,  the  openings  between 
the  columns  are  increased,  and  objects  are  seen  beyond 
which  in  point  of  fact  would  be  hidden.  This  dis- 
crepancy is  not  very  important  and  in  general  would 
hardly  be  noticed,  but  the  altered  relations  between 
these  circular  figures  and  other  objects  in  their  imme- 
diate neighborhood  is  a  more  serious  matter.  The 
square  abacus  between  the  shaft  and  the  sphere  that 
surmounts  it  looks  too  big  for  its  place  if  left  without 
correction,  and  looks  smaller  than  its  fellows  if  reduced 
as  the  sphere  and  cylinder  are.  So  also  when  an  octa- 
gon occurs  in  immediate  connection  with  a  circle.  If  its 
shape  is  adjusted  to  that  of  the  corrected  ellipse,  its 
want  of  harmony  with  the  rest  of  the  drawing  often 
becomes  painfully  apparent.  A  satisfactory  adjustment 
may  sometimes  be  effected  by  a  compromise,  the  ellipse 
being  made  not  quite  horizontal  and   the  octagon  or 


164  MODERN   PERSPECTIVE. 

square  being  not  quite  harmonized  with  it.  But  a  per- 
fectly satisfactory  adjustment  is  in  some  of  these  cases, 
and  notably  in  the  case  of  a  row  of  columns,  almost 
impossible. 

These  difficulties  are  of  course  greater  as  the  objects 
in  question  are  further  removed  from  the  centre  of  the 
picture,  and  may  be  diminished  or  removed  altogether 
by  so  taking  the  position  of  the  picture  and  that  of 
the  spectator  that  the  circular  object  is  at  or  near  the 
centre  V^. 

260.  Although  the  distortions  of  circular  and  spheri- 
Distortion  of  ^^^  objccts  are,  in  general,  the  only  ones  that 
all  figures.  ^^^^  loudly  for  corrcction,  and  it  is  to  them 
alone  that  correction  is  systematically  applied,  it  is  ob- 
vious that  a  similar  distortion  must  exist  for  all  objects 
equally  distant  from  the  centre  of  the  picture,  the  so- 
called  distortion  consisting  in  this,  that  the  shape  of  the 
object  in  the  drawing  is  different  from  the  shape  which  the 
real  object  presents  to  the  eye  (224,  225).  This  is  in  fact 
implied  in  the  fundamental  principle  of  perspective,  the 
principle  that  a  perspective  drawing  will  look  right  from 
only  one  point,  namely,  the  station-point.  Now  as  from 
the  station-point  every  part  of  the  picture  except  the 
centre  is  viewed  obliquely,  askance,  as  it  were,  every- 
thing must  be  drawn  of  a  different  shape  from  wliat 
it  appears  in  order  that  when  the  drawing  is  looked  at 
thus  obliquely  it  may  appear  as  the  object  itself  does 
when  looked  at  directly.  By  the  very  theory  of  per- 
spective only  the  object  just  opposite   the   eye,  seen 


THE   HUMAN   FIGURE.  165 

along  the  axis  of  the  picture,  just  at  its  centre,  is  drawn 
as  it  looks.  Everything  else  is,  so  to  speak,  distorted. 
The  outline  given  to  it  is  not  its  real  outline,  but  one 
which  will  look  like  its  real  outline  when  seen  side- 
ways from  the  position  assigned  to  the  spectator.  This 
distortion  is  inevitable,  and  every  object  in  a  perspec- 
tive drawing,  except  the  one  at  the  centre,  is  always 
distorted. 

This  is  most  noticeable  when  objects  are  drawn  in  par- 
allel perspective,  as,  for  example,  in  Fig.  23,  Plate  YI. 
But  all  objects  are  more  or  less  distorted  and  exag- 
gerated in  size.  They  are  stretched  out  in  a  direction 
away  from  the  centre  of  the  picture,  just  like  the 
shadows  in  Fig.  60. 

261.  The  disfigurement  produced  by  this  does  not 
of  course  become  very  obvious,  except  for  circles  and 
spheres,  so  long  as  objects  are  not  far  removed  from  the 
centre,  that  is  to  say,  so  long  as  the  picture  is  of  moder- 
ate extent.  The  limit  commonly  assigned  to  a  persp^- 
tive  drawing  is  sixty  degrees,  that  is  to  say,  the  width 
of  the  picture  should  not  be  greater  than  its  distance 
from  the  station  point.  But  this  implies  that  the  centre 
V^  is  in  the  middle  of  the  picture,  which  as  we  have  seen 
is  often  not  the  case,  and  it  is  better  to  say  that  no  part 
of  the  picture  should  be  distant  from  the  point  opposite 
the  eye  more  than  half  the  distance  of  the  spectator  in 
front  of  it.  But  even  within  this  range  the  distortion 
even  of  rectilinear  objects  is  sometimes  intolerable,  and 
great  caution  must  always  be  used  in  regard  to  objects 
situated  at  the  edges  of  the  picture. 


166  MODERN   PERSPECTIVE. 

262.  This  limit  of  sixty  degrees  is  obviously  an 
The  extent  of  arbitrary  one,   and   only   means  that   by  the 

the  range  of      .  .      . 

the  picture,  time  it  IS  reached  the  distortion  begins  to  be 
noticeable.  It  is  foolish  to  say,  as  is  sometimes  said, 
that  this  is  fixed  because  sixty  degrees  embrace  all  that 
one  can  see  without  turning  his  eyes,  or  as  others  say, 
without  turning  his  head,  and  that  this  is  accordingly 
the  natural  range  for  a  picture.  For  one  has  to  turn  his 
eyes,  more  or  less,  to  see  directly  anything  larger  than  a 
pin-head,  held  at  arm's  length ;  and  he  has  no  need  to 
turn  his  head  even  to  embrace  a  horizon  of  ninety  de- 
grees. Besides,  why  should  not  one  turn  his  head  as 
well  as  his  eyes  in  looking  at  a  picture  as  well  as  in 
looking  at  nature  ?  If  he  is  at  the  station-point,  where 
he  ought  to  be,  turning  his  head  cannot  make  things 
look  wrong,  and  if  he  is  not  there  keeping  it  still  will 
not  make  them  look  right. 

263.  There  is,  nevertheless,  a  remarkable  difference 
between  turning  the  head  to  look  at  a  thing,  and 
merely  turning  the  eye.  The  plane  of  the  picture,  so 
to  speak,  regarding  the  aspect  of  nature  as  a  picture, 
is  conceived  to  be  parallel  to  one's  face.  Turning  the 
head  seems  to  alter  the  position  of  the  plane  of  projec- 
tion. If  one  looks  straight  down  a  street  he  seems  to 
see  it  in  parallel  perspective.  If,  keeping  his  eye  fixed 
on  the  same  spot,  he  turns  his  head,  the  street  seems 
now  to  be  in  angular  perspective,  though  the  image  on 
the  retina  has  not  been  disturbed.  The  right-angles 
will  seem  to  become  acute  and  obtuse.  A  horizontal 
circle  on  the  floor  or  ceiling  will  look  like  an  ellipse 


THE   HUMAN    FIGURE.  167 

with  a  horizontal  axis  if  you  face  it.  Turn  your  head 
away,  without  moving  your  eyes,  and  though  it  really 
looks  just  as  it  did  before,  being  regarded  from  the  same 
point,  its  axes  will  seem  inclined. 

264  What  has  been  said  of  cylinders  and  spheres 
strictly  applies  to  the  human  figure,  which  t^^^  j^^^^^j^ 
may  be  regarded,  in  a  rough  way,  as  a  cylin-  ^°'^''^' 
der  surmounted  by  a  sphere.  Perspective  distortion  is 
here  even  more  intolerable  than  in  the  case  of  the  more 
exact  geometrical  solids,  and  the  need  of  correction  is 
more  imperative. 

This  is  excellently  illustrated  by  the  phenomena  of 
the  familiar  parlor  amusement  called  "Chinese  q^^^^^^^ 
Shadows,"  in  which  a  sheet  is  hung  across  the  ^^^'^''''^^ 
middle  of  a  room  and  the  shadows  of  the  performers  on 
one  side  are  thrown  upon  it  for  the  entertainment  of 
spectators  on  the  other  side.  A  single  lamp  is  used, 
and  it  is  obvious  that  all  the  shadows  except  the  one 
just  opposite  the  lamp  and  on  a  level  with  it,  must  be 
more  or  less  distorted  as  they  are  more  or  less  removed 
from  this  centre.  But  it  is  also  obvious  that  if  one  of 
the  spectators  places  himself  exactly  opposite  the  lamp 
and  as  far  in  front  of  the  screen  as  the  light  is  behind 
it,  the  distorted  outlines  will  be  foreshortened  into  the 
true  shape  of  the  figures  on  the  other  side  as  seen  from 
the  place  occupied  by  the  flame. 

265.    An  historical  picture  then,  if  painted  in  true 
perspective,  with  all  its  figures  so  drawn  as  Historical 
to  present  their  true  aspect  to  the  spectator  p^^*"^^"- 


168  MODERN   PEKSPECTIVE. 

standing  at  a  given  point  in  front  of  it,  would  have* all 

its  personages  as  much  out  of  drawing  as  a're  Chinese 

Shadows  upon  a  screen.    Fig.  60,  which  exhib- 

Fig.  60.  .  ,  \  ,> 

its  the  results  ot  an  experiment  with  a  group 
of  statuary  and  with  half  a  dozen  round  balls,  illustrates 
these  conclusions.  They  are,  as  happens  in  perspective 
to  all  objects  (260),  stretched  out,  in  a  direction  away 
from  the  centre,  Y^. 

266.  No  such  picture  of  course  was  ever  painted, 
painters  always  adopting  the  same  course  for  figures 
that  has  been  recommended  for  their  geometrical  pro- 
totypes. Every  figure  is  outlined  independently  of  all 
the  others,  and  in  its  natural  proportions,  just  as  if  it 
occupied  the  centre  of  the  picture.  In  order  to  see  it 
correctly  the  spectator  has  to  stand  opposite  to  it, 
and  as  he  cannot  of  course  stand  exactly  opposite  to 
more  than  one  figure  at  a  time,  it  follows  that  he  can 
never  see  more  than  one  at  a  time  in  correct  drawing. 
All  the  others  are  distorted  by  foreshortening.  But  if 
he  is  at  a  distance  from  the  picture  this  distortion  is 
not  noticeable,  and  when  he  comes  near  he  confines  his 
attention  to  the  figure  nearest  his  eye. 

This  is  very  well  illustrated  by  Eaffaelle's  fresco 
Fig.  63.  called  the  ''  School  of  Athens,"  of  which  the 
of  Athens,  principal  part  is  sliown  in  Fig.  63.  The 
figures  on  the  extreme  right  and  the  spheres  which 
tliey  carry  are  drawn,  as  has  been  pointed  out  by  M. 
Thiebaut,  just  as  if  they  were  in  the  centre  of  the  pic- 
ture, opposite  the  eye.  If  the  spectator  were  in  the 
room,  one  wall  of  which  is  occupied  by  this  picture,  the 


THE   HUMAN   FIGURE.  169 

chief  one  of  these  figures  would  appear  foreshortened,  as 
in  Fig.  64,  a,  the  sphere  looking  like  an  Qgg 

^  .  .         ,  Fig-  64,  a. 

on  end.  One  sometimes  m  the  scenery  of  a 
theatre  sees  a  round  ball  on  a  post  present  a  similar 
aspect,  from  the  same  cause.  In  order  to  make  the  fig- 
ure appear  as  it  should,  to  make  it  assume  when  seen 
from  the  middle  of  the  room  the  form  intended  by  the 
painter,  he  would  have  had  to  draw  it  as  shown 

^  Fig.  64,  b. 

in  Fig.   64,  &,  giving  the  sphere    a  flattened 

form,  just  as  in  Fig.  59  c. 

267.    If  his  picture  is  a  large  one,  the  painter  often 

has  a  difficult  task  to  reconcile  his  background  The  architec- 
tural baek- 
and  accessories,  which  are  drawn  according  to  ground. 

perspective  rule,  and  calculated  to  be  seen  from  a  single 
point,  with  his  figures,  which  utterly  violate  these  rules, 
and  permit  and  indeed  require  the  spectator  to  regard 
them  from  half  a  dozen  different  positions.  The  back- 
ground proper  may  not  give  much  trouble.  More  of 
it  will  be  seen  between  the  figures  than  would  be  the 
case  in  nature,  as  has  been  pointed  out  already  in  re- 
gard to  columns,  and  care  must  be  taken  not  to  use  any 
forms  which  require  the  spectator  to  remain  exactly  at 
the  station  point ;  for  this  he  will  not  do.  But  that  can 
easily  be  managed.  The  chief  difficulty  is  found  in 
fitting  the  figures  into  the  foreground.  If  a  chair,  for 
instance,  occupies  one  end  of  the  front  of  the  picture 
and  is  put  into  perspective  along  with  the  walls  and 
floor,  so  as  to  appear  correctly  from  a  point  opposite  the 
middle  of  the  picture,  it  must  needs  look  more  or  less 
crooked  when  looked  at  from  a  point  opposite  the  end 


170  MODERN   PERSPECTIVE. 

of  the  picture,  and  it  is  no  easy  matter  to  make  a  figure 
painted  from  that  point  of  view  look  as  if  lie  were  seated 
comfortably  in  it.  If,  moreover,  he  is  to  be  represented 
as  looking  straight  across  the  picture,  it  is  by  no  means 
clear  whether  he  should  be  drawn  in  profile,  as  he  would 
appear  from  tliis  point,  or  with  the  three-quarter  face 
which  he  would  show  from  the  other. 

268.  There  is  even  greater  difficulty  in  reconciling  the 
perspective  of  a  floor  witli  the  want  of  perspective  of  the 
feet  that  stand  upon  it.  If  a  number  of  persons  are  shown 
in  the  foreground  of  a  picture  all  facing  the  same  way, 
it  is  impossible  to  make  the  direction  of  their  feet  agree 

with  that  of  the  boards  on  which  they  stand. 

Fig.  61,  a.  "^ 

Fig.  61,  a,  shows  how  the  feet  of  a  dramatic 
company,  seen  just  as  the  curtain  is  descending,  would 
be  drawn  in  true  perspective,  agreeably  to  the  perspec- 
tive plan  below.      Fig.   62   sliows  the  neces- 

Fig.  62. 

sary  correction,  each  pair  of  feet  being  drawn 
just  as  if  it  were  exactly  opposite  the  spectator.  But 
it  is  to  be  noticed  that  the  end  man  is  necessarily 
represented  as  standing  diagonally  across  the  floor 
boards. 

269.  In  general,  the  attitude  of  the  figures  at  the 
edge  of  a  large  picture  is  not  very  clearly  defined,  and 
varies  as  the  spectator  changes  his  position.  In  Guido's 
"  Aurora,"  for  instance,  if  one  stands  opposite  one  end 
of  the  picture  the  figure  at  the  extreme  left  seems  to  be 
marching  along  the  front,  and  Aurora  herself  to  be  look- 
ing out  of  the  frame.  If  he  goes  to  the  other  end  she 
seems  to  be  looking  back  at  Apollo  in  the  chariot,  and 


THE    HUMAN   FIGURE.  171 

the  other  figure  seems  to  be  just  coming  round  the  cor- 
ner, so  to  speak,  from  behind  his  car.  The  analogous 
phenomenon  of  a  portrait  seeming  to  follow  one  with  its 
eyes  is  commonly  observed.  It  is  not  so  frequently  re- 
cognized that  the  whole  face  seems  to  turn,  especially 
when  a  front  face  is  shown,  the  cheek,  not  the  nose, 
seeming  to  be  the  most  prominent  feature. 

In  the  upper  part  of  very  high  pictures  the  heads 
have  to  be  drawn  as  they  appear  when  seen  from  be- 
neath, the  under  side  of  the  jaw  and  of  the  eyebrows 
being  shown.  But  this  makes  the  head  seem  to  fall 
backwards  as  one  recedes  from  it.  In  Titian's  "As- 
sumption," for  example,  the  attitude  of  the  head  va- 
ries greatly  according  to  the  position  of  the  spectator. 
From  the  opposite  end  of  the  room  in  which  it  hangs, 
the  face  seems  to  be  turned  up  and  the  head  thrown 
back.  As  one  approaches  the  picture  it  seems  to  bow 
forward. 

Another  difficulty  in  the  perspective  of  figure-subjects 
is,  that  if  the  figures  are  as  large  as  life,  everything  nearer 
than  they  are  becomes  colossal.  But  this  may  be  avoided, 
and  generally  is,  by  not  having  anything  in  particular 
in  front  of  the  principal  figures. 

270.  All  the  difficulties  encountered  with  the  human 
figure  are  met  in  even  greater  force  with  figures  of  ani- 
mals, except  that  it  is  possible  for  them  to  be  consider- 
ably out  of  drawing  without  detection.  Fig. 
65,  which  is  borrowed  from  the  work  ot  M. 
Thiebaut,  illustrates  at  once  the  extent  of  the  distortion 
and   the  difficulty  of  correcting  it.     The  form  of  the 


172  MODERN   PERSPECTIVE. 

pedestal  controls  the  position  of  the  horse's  hind  leers, 
and  necessitates  a  distortion  for  which  there  seems  no 
remedy.  A  distortion  which  is  hardly  noticed  in  a  four- 
legged  table  is  intolerable  in  a  quadruped.  They  should 
never  be  drawn  in  parallel  perspective. 


PHOEBE  A.    HEARST 
ARCHITECTimAL  LIBRARY 


CHAPTER  XIII. 

CYLINDEICAL,  CURVILINEAR,   OR  PANORAMIC   PERSPECTIVE. 

271.  The  previous  chapter  has  discussed  the  so-called 
distortions  to  which  circular,  cylindrical,  and  spherical  ob- 
jects are  subjected  when  drawn  according  to  the  methods 
of  plane  perspective,  and  has  explained  the  so-called 
corrections  which  are  applied  to  such  objects.  Similar 
distortions,  it  was  shown,  attend  the  putting  of  the 
human  figure  and  animals  into  perspective,  and  similar 
corrections  apply.  Indeed,  it  was  pointed  out  (260) 
that  every  object  not  exactly  at  the  centre  of  tlie  picture 
must  necessarily  be  more  or  less  out  of  drawing,  though 
the  distortion  is  not  generally  such  as  to  attract  notice 
save  in  the  cases  mentioned. 

272.  Plate  XV.  demonstrates  the  existence  of  these 
distortions,  exhibits  some  instances  in  which  piatexv. 
they  are  intolerable,  even  in  the  case  of  recti-  Distortions 

of  rectilinear 

linear  objects,  and  shows  yet  another  way  of  objects, 
correcting  them.  By  distortion,  as  has  been  said,  we 
mean  that  the  outline  given  in  the  drawing  is  different 
from  the  outline  presented  to  the  eye  by  the  object 
drawn.  Now  the  rays  of  light  that  pass  from  the  out- 
line of  an  object  to  the  eye  form  an  irregular  cone,  whose 
base  is  this  outline  itself  (255).     The  perspective  repre- 


174  MODERN   PERSPECTIVE. 

sentation  of  this  outline  is  the  line  in  which  this  cone  of 
rays  is  cut  by  the  plane  of  the  picture.  If  this  plane  cuts 
the  cone  of  rays  in  a  direction  at  right  angles  to  its  axis 

—  that  is  to  say,  if  the  object  is  at  the  centre  of  the  pic- 
ture —  then  the  section  is  of  the  same  shape  as  the  base  ; 
the  perspective  is  of  the  same  shape  as  the  object.  But 
if  the  plane  of  the  picture  cuts  the  cone  of  rays  obliquely 

—  as  must  be  the  case  with  all  objects  not  just  at  the 
centre  —  then  the  section  is  not  of  the  same  shape  as  the 
base,  and  the  perspective  does  not  look  like  the  object; 
it  is,  so  to  speak,  distorted.  Of  course,  when  seen  from 
the  station  point,  obliquely,  the  perspective  is  fore- 
shortened,  and  looks  just  as  the  object  does.  But  in  it- 
self, and  when  looked  at  merely  as  a  line,  it  presents  a 
different  form  (224,  225,  256,'  260). 

273.  This  is  illustrated  in  Fig.  70,  in  which  is  seen,  at  h, 
a  rectangular  block,  drawn  in  parallel  perspec- 
tive, but  considerably  to  the  right  of  the  centre. 
Its  aspect  is  such  as  no  rectangular  block  could  ever 
possibly  present  to  the  eye.  It  exhibits  three  faces, 
one  of  which  is  a  square.  But  if  a  rectangular  block  is 
held  so  that  one  of  its  faces  shows  four  right  angles,  it 
must  be  held  so  that  neither  of  the  other  faces  can  be 
seen  at  all.  If,  on  the  other  hand,  it  stands  so  that  all 
three  faces  are  seen,  as  this  block  evidently  does,  then 
all  the  angles  must  appear  either  acute  or  obtuse.  The 
figure  within  the  circle  shows  how  such  a  block  really 
looks  when  one  looks  straight  at  it,  and  this  is  the  way  it 
is  drawn  when  at  the  centre  of  the  picture.  The  differ- 
ence between  these  two  representations  exemplifies  the 


Fig.  70.  ° 


CYLI^'DRICAL   PERSPECTIVE.  175 

distortion  to  which  all  shapes  are  subjected  when  the 
line  from  the  object  to  the  eye  is  not  at  right  angles  to 
the  picture. 

But  this  distortion  in  the  drawing  is  corrected,  by 
foreshortenino-  when  one  looks  at  the  drawing-  ^.    ^^ 

^'  ^    Fig.  70,  a. 

from  the  station  point,  S,  which  in  this  case  is 

P    .  Fig.  70,  6. 

a  few  inches  in  front  of  V  ,  in  Fig.  66.     In 

fact,  Fig.   70,  a,  was  sketched  from  this  point,  and  is 

a  view,  not  of  the  cube  itself,  but  of  the  Fig.  70,  h, 

thus  foreshortened  into  a  real  likeness  of  the  object  it 

represents. 

274.  Fig.  66  exhibits  other  and  even  more  striking 
phenomena.     Take  first  the  church  on  the  left 

A  street. 

hand.  It  is  horribly  out  of  drawing,  although 
the  picture  does  not  extend,  on  this  side,  very  far  from 
the  centre.  Not  only  is  the  church  and  the  belfry 
twisted  out  of  shape,  but  their  dimensions  increase 
instead  of  diminishing  as  they  recede  from  the  eye. 
This  sort  of  distortion  is  often  seen  in  old-fashioned 
prints,  and  in  photographs  of  very  long  buildings,  or  of 
interiors,  taken  nearly  in  elevation.  It  arises,  as  is 
obvious,  from  both  vanishing  points,  V^  and  V^,  being 
on  the  same  side  of  the  object. 

275.  It  is  accordingly  a  maxim  in  perspective  that 
the  objects  represented  must  lie  between  their  principal 
vanishing  points,  and  that  the  plane  of  the  picture  must 
be  so  taken  as  to  effect  this.  In  other  words,  the  angu- 
lar range  of  the  picture  to  the  left  of  the  center,  Y*^,  must 
not  exceed  the  angle  between  the  plane  of  the  picture 
and  the  right-hand  side  of  the  object,  and  vice  verscv 


176  MODERN   PEKSPECTIVE. 

In  the  figure,  for  example,  the  front  line  of  the  church 
and  houses  makes  an  angle  of  about  20°  with  the  picture. 
V^  is  accordingly  about  20°  to  the  left  of  C,  and  that  is 
as  far  as  the  picture  of  objects  whose  lines  are  directed 
to  V^  and  V^,  like  these,  can  be  extended  in  that 
direction. 

276.  If  then  it  is  desired  to  embrace  a  considerable 
extent  of  horizon,  and  at  the  same  time  to  represent 
objects  as  being  nearly  parallel  to  the  picture,  there  is 
no  choice  but  to  take  the  plane  of  the  picture  in  such  a 
way  that  they  shall  be  exactly  parallel.  That  is  to  say, 
Parallel  Perspective  must  be  employed.  For  where  a 
rectangular  object  has  one  of  its  sides  very  nearly  par- 
allel to  the  picture  the  horizontal  lines  of  the  other  side 
must  be  nearly  perpendicular  to  it,  and  their  vanish- 
ing point  very  near  the  centre,  V^.  Unless,  then,  the 
centre  is  set  quite  at  the  edge  of  the  picture,  which  is 
undesirable,  the  distortion  shown  in  the  figure  must 
occur. 

If,  for  example,  the  opposite  sides  of  a  room  are 
both  to  be  shown  at  once,  it  will  not  do  to  set 

Interiors.  i  •  i       i  • 

the  end  of  the  room  at  an  angle  with  the  pic- 
ture, however  acute ;  for  the  end  wall  will  contain  the 
vanishing  point  of  lines  parallel  to  the  sides,  and  one  of 
the  side  walls  that  of  lines  parallel  to  the  end.  The 
other  side  will  then  be  beyond  both  vanishing  points, 
and  must  experience  this  disagreeable  twisting,  as  is 
often  seen  in  paintings  and  photographs.  It 
is  the  same  with  street  scenes.  If  both  sides 
of  a  street  are  to  be  seen  at  once,  they  must  be  perpen- 


CYLINDRICAL  PERSPECTIVE.  177 

dicular  to  the  plane  of  the  picture.     Fig.  66  sufficiently 
shows  what  will  happen  if  they  are  not. 

277.  The  distortions  at  the  other  end  of  the  pic- 
ture, however,  though  less  offensive,  and  consequently 
much  more  common,  are  almost  as  great.  The  buildings 
are  too  long  and  the  hills  not  steep  enough.  Both  have 
quite  different  proportions  from  what  they  would  pre- 
sent to  a  spectator  at  S,  and  are  quite  unlike  any  sketch 
that  would  be  made  of  them  from  that  point.  For  the 
proportions  of  an  object,  that  is  to  say,  the  relative  size 
of  its  parts,  depend  upon  the  relative  angular  dimension 
of  the  parts,  that  is,  upon  the  relative  size  of  the  angles 
they  subtend  at  the  eye.  The  apparent  distance  apart 
of  the  points  that  define  them  right  and  left,  or  up  and 
down,  is  angular  distance.  A  painter,  then,  Angular  di- 
who  would  represent  things  in  their  true  pro-  "^*'"^'°°^- 
portions,  as  they  look,  and  in  their  apparent  relations 
one  to  another,  would  have  to  proportion  the  linear 
dimensions  upon  his  canvas  to  the  angular  dimensions 
of  his  object.  And  this,  in  fact,  is  just  what  every 
painter,  every  draughtsman  of  whatever  kind,  always 
does  when  he  undertakes  to  sketch  from  nature.  It 
is  the  method  of  every  artist  who  undertakes,  outdoors 
or  in,  to  draw  things  as  he  sees  them ;  he  can  have  no 
other  ;  he  must  give  to  the  representation  of  objects  the 
apparent  shape  and  the  relative  size  that  the  objects 
themselves  present  to  his  eye.  In  other  words,  he  pro- 
portions the  linear  dimension  upon  his  canvas  to  the 
angular  dimension  of  the  object.  But  this  is  exactly 
what  perspective  does  not  do.     In  sketching,  one  may 

12 


178  MODERN   PERSPECTIVE. 

begin  in  the  middle,  fix  the  position  of  his  central  ob- 
ject, and  he  will  naturally  distribute  other  things  about 
it  to  the  right  and  left,  according  to  their  apparent  dis- 
tance from  it.  He  proportions  their  distance  to  their 
anovular  distance,  and  their  size  to  their  angular  dimen- 
sions ;  that  is,  to  the  difference  of  the  angular  distance 
of  their  edges.  But  in  a  perspective  drawing,  as  is 
clearly  shown  in  the  plan,  the  distance  of  an  object 
from  the  centre  of  the  picture  is  not  proportional  to  its 
apparent,  or  angular,  distance,  but  is  constantly  greater, 
being  proportionate  to  the  tangent  of  the  angle,  and  its 
size  is  accordingly  proportioned,  not  to  its  angular  di- 
mension, but  to  the  difference  of  the  tangents  of  the 
angular  distance  of  its  edges  from  the  centre.  The  scale 
to  which  they  are  drawn  accordingly  increases  from  the 
centre  outward,  just  as  in  Mercator's  Projection,  which 
gives,  indeed,  a  sort  of  perspective  view  of  the  terrestrial 
sphere,  as  seen  from  a  station  point  at  its  centre. 

278.  In  view  of  these  evils,  and  of  the  practical  im- 
possibility of  escaping  them  by  using  the  station  point, 
the  attempt  has  been  made  to  avoid  them  by  represent- 
ing objects  just  as  they  appear,  making  the  linear  di- 
mensions in  the  drawing  proportional  to  the  apparent  or 
angidar  dimensions  in  nature.  It  is  plain  that  this 
scheme  could  be  thoronghly  carried  out  only  by  draw- 
ing on  the  inside  of  a  hollow  spherical  surface,  a  condi- 
tion impossible  to  fulfil.  A  cylindrical  surface,  however, 
answers  nearly  as  well,  especially  when,  as  is  usually  the 
case,  the  vertical  dimensions  are  relatively  small.  A 
cylinder,  moreover,  has  the  advantage  of  being  a  deveU 


CYLINDRICAL   PERSPECTIVE.  179 

ojpable  surface;  it  can  be  rolled  out  flat.  This  is  the  sur- 
face employed  in  circular  panoramas,  and  it  is  virtually 
that  employed  in  sketching  from  nature.  For  as  one 
turns  from  one  object  to  another  he  virtually  keeps  the 
corresponding  part  of  his  canvas  directly  in  front  of 
him,  at  right  angles  to  his  line  of  vision,  just  where 
a  cylindrical  surface  would  be. 

279.  The  plate  illustrates  the  result  of  this  procedure, 
and  affords  an  opportunity  of  comparino-  it  with   Projection 

^^  "^  1  o  uponacylin- 

the  results  of  plane  perspective.     In  the  plan  der,  instead 

i-  i^         c  JT  of  upon  a 

of  the  street  we  have  the  position  of  the  spec-  p^^''*'- 
tator  indicated  at  S ;  that  of  a  transparent  plane,  ^'^^-  ^'  *"' ■ 
representing  a  picture  plane,  at  p  V^/>,  and  that  of  a 
transparent  cylinder  at  a  Y^  h.  The  centre  of  the  per- 
spective picture  is  at  V^,  the  point  nearest  the  specta- 
tor, and  the  plane  and  cylinder  are  tangent  at  that  point. 
Visual  rays  drawn  from  the  principal  points  in  the  street 
to  the  station  j)oint  pierce  both  surfaces,  and  pictures 
drawn  upon  them  would,  when  seen  from  the  point  S, 
obviously  coincide  with  each  other  and  exactly  cover 
the  objects  represented. 

280.  Fig.  66  exhibits  the  result,  as  shown  on  the 
plane  pp,  and  Fig.  67  that  shown  on  the  cylinder  a  b. 
The  first  strikingly  illustrates  what  has  been  said  of  the 
inevitable  distortion  of  objects  in  plane  perspective,  and 
of  their  gradual  exaggeration  of  scale  as  they  recede 
from  the  centre.  Fig.  67  shows  the  effect  of  making 
the  linear  dimensions  in  the  drawing  correspond  to  the 
angular  dimensions  of  the  objects  drawn,  that  is  to  say, 
of  drawing  everything  just  as  it  appears.      Of  these 


180  MODERN   PERSPECTIVE. 

effects  the  most  noticeable  are  these  :  that  in  the  first 
place  the  distortion  of  the  church  on  the  left  entirely 
disappears,  and  in  the  second  place  the  distortion  on  the 
right  disappears  also,  the  houses  and  the  landscape  be- 
yond being  reduced  to  dimensions  proportioned  to  the 
dimensions  given  to  the  nearer  objects,  while  the  size  of 
the  picture  is  greatly  diminished.  All  this  is  a  great 
gain.  But  on  the  other  hand  the  horizontal  parallel 
lines  are  all  more  or  less  curved.  The  lines  which  in 
Fig.   66   are  all   straight   and  converG^e  to  a 

straight  .  ^  ° 

lines  drawn     siuole  vauishins^   point,  in   Ym.   67  convero-e 

as  curves  *-  o     i  '  O  to 

towards  these  two  vanishing  points.  Indeed, 
every  horizontal  line,  except  those  in  the  plane  pass- 
ing througli  the  eye,  is  drawn  as  curved.  Its  per- 
spective lies  in  the  line  in  which  the  plane  passing 
through  the  eye  and  the  line  itself  intersects  the  cylin- 
der. This  is  an  ellipse  in  space,  and  its  development  is 
a  curved  line,  concave  towards  the  horizon,  which  it 
crosses  at  points  180°  distant  from  each  otlrer,  the  per- 
spective of  its  vanishing  points.  This  curvature  would 
of  course  disappear  if  the  paper  were  bent  into  a  cylin- 
drical form,  and  the  eye  placed  at  the  axis  of  the  cylin- 
der, opposite  the  horizon,  and  in  the  large  circular  pano- 
ramas which  are  sometimes  exhibited,  and  which  have 
given  to  this  method  the  name  of  Panoramic  Perspec- 
tive, this  of  course  is  done.  But  in  general  the  de- 
veloped cylinder  has  to  remain  flat,  and  it  must  be 
confessed  that  this  curvature  of  lines,  which  in  nature 
are  straight,  is  itself  a  distortion  which  most  persons 
find  extremely  objectionable. 


CYLINDEICAL   PERSPECTIVE.  181 

281.  It  is  worth  while  to  remark,  however,  that 
this  phenomenon  of  the  apparent  curvature  of  straight 

.  .         lines  often 

straight   hues   is   oi   constant   occurrence    m   seem  to  be 

"^  curved  in 

nature  ;  and  it  is  just  one  of  those  phenomena  mature, 
of  nature  with  which  perspective  has  to  do,  being  con- 
cerned with  the  appearances  of  parallel  lines.  All  sys- 
tems of  lines  which  are  long  enougli  to  indicate  both 
their  vanishing  points,  converging  to  one  point  on  the 
right  and  to  another  on  the  left,  have  an  apparent  cur- 
vature. Such,  as  has  already  been  pointed  out,  are  the 
long  parallel  lines  of  cloud  which  often  cover  the  sky, 
or  the  sunbeams  and  shadows  which  sometimes  at  sun- 
set pass  completely  over  from  west  to  east.  In  both 
these  cases  each  particular  cloud  or  sunbeam,  as  one 
looks  at  it,  seems  quite  straight ;  but  all  the  others  on 
either  side  seem  concave  towards  it.  In  fact,  as  they  all 
meet,  or  tend  to  meet,  at  two  different  points,  and  to 
separate  between  them,  they  miist  seem  curved  ;  straight 
lines  can  meet  at  only  one  point. 

It  is  the  same  with  the  horizon  itself,  which  seems 
straight  when  one  looks  at  it,  but  seems  curved  when 
one  looks  up  or  dow^n.  So  with  other  long  lines,  such 
as  eaves,  sidewalks,  or  housetops.  As  one  turns  his  eye 
rapidly  from  one  end  of  a  street  to  the  other,  the  appar- 
ent curvature  reveals  itself  unmistakably. 

Now  as  all  straight  lines  in  nature,  if  prolonged,  seem 
to  be  curved,  approaching  the  horizon  at  the  gradually 
increasing  angle,  so  in  Panoramic  Perspective,  their 
perspective  representations  do  curve,  until  they  cut 
the  horizon. 


182  MODERN   PERSPECTIVE. 

282.  To  one  who  is  accustomed  to  observe  this  curious 
phenomenon,  the  curvature  of  the  lines  in  Cylindrical,  or, 
as  we  may  now  call  it,  Curvilinear,  Perspective  is  but  a 
trifling  evil,  hardly  to  be  counted  against  its  manifold 
advantages.  Of  these  the  chief  is  perhaps,  as  has  been 
said,  the  perfect  conformity  of  its  results  with  those  ob- 
tained in  sketching  from  nature.     Of  this  an  excellent 

illustration  is  afforded  by  Fiw.  68,  a  rude  out- 
Fig  68.       .  J        &         ' 

line  sketch  from  a  water-color  by  Turner,  rep- 
resenting the  Ducal  Palace  at  Venice,  and  the  adjacent 
buildings.  He  sketched  each  building  just  as  it  looked, 
and  did  not  mind  the  resulting  curvature  of  the  horizon- 
tal lines  of  his  drawing. 

This  drawing  exhibits,  however,  what  is  perhaps  as 
objectionable  a  distortion  as  any,  an  apparent  convex- 
ity in  the  objects  represented.  The  quay,  which  in  fact 
is  straight,  looks  convex,  as  does  also  the  village  street 
in  Fig.  67. 

In  photographs  taken  with  a  revolving  camera,  the 
pictures  are  virtually  taken  upon  a  cylindri- 
cal surface,  and  exhibit  the  same  phenomena. 
Photographs  taken  with  a  common  camera  are  in  Plane 
Perspective. 

283.  This  convexity  is  not,  however,  very  noticeable, 
except  in  cases  like  these,  where  a  long  line  is  parallel 
to  the  cylinder  near  the  middle  of  the  picture,  its  per- 
spective being  horizontal  in  the  middle  and  curving  to 
the  right  and  left.  Where  the  subject  is  so  chosen  that 
horizontal  lines  occur  only  at  the  ends  of  the  picture, 
so  that  the  lines  curve  only  one  way,  both  the  curvature 


CYLINDRICAL   PERSPECTIVE.  183 

and  the  convexity  are  less  noticeable;   and  they  can 
hardly  be  detected  where  the  lines  are  short  and  broken. 
In  such  cases  the  special  characteristic  of  Cur-  ^j^^  ^^^^^^^ 
vilinear  or  Panoramic  Perspective,  that  it  per-  raiTge  onL 
mits  the  limits  of  the  picture  to  be  extended  ^'*^'"''®' 
indefinitely  without  the  rapidly  increasing  distortions  to 
which   Plane  Perspective  is  liable,  can  be  taken  full 
advantage  of.     A  drawing  in   Curvilinear  Perspective 
may  often  be  made  to  embrace  a  hundred,  or  even  a 
hundred  and  twenty  degrees  of  horizon,  with  less  embar- 
rassment than  is  in  Plane  Perspective  incurred  by  sixty. 
Fig.  69,  which  is  borrowed,  though  much  re- 

*  '  o  Fig.  69. 

duced,  from  a  rare  and  little-known  work,  by 
Mr.  W.  G.  Herdman,  published  in  Liverpool  in  1853, 
exhibits  this  excellence  in  a  striking  degree.  It  repre- 
sents the  meeting  of  two  streets  in  some  foreign  town, 
and  succeeds  in  showing  both  sides  of  both  streets, 
without  distorting  any  part  of  either.  The  horizontal 
anole  embraced  must  be  more  than  a  hundred  decrees. 
Most  of  the  dotted  curved  lines,  which  in  the  original 
were  carried  across  the  picture  in  order  to  show  the 
theory  on  which  the  drawing  is  constructed,  are  here 
omitted.  Where,  as  here,  the  sky-lines  are  broken,  the 
principal  objects  in  angular,  not  in  parallel  perspective, 
and  the  continuous  horizontal  lines  few,  the  disadvan- 
tages of  this  method,  as  is  evident  from  the  figure,  are 
reduced  to  a  minimum. 

284.  Whether  in  any  given  case  plane  or  cylindrical 
perspective  is  to  be  preferred  is  a  matter  of  judgment, 
and  one's  decision  must  depend  chiefly  upon  the  nature 


184  MODERN   PERSPECTIVE. 

of  his  subject.  For  architecture,  except  in  picturesque 
sketches,  the  latter  is  in  general  obviously  unfit;  but 
for  the  landscape  painter  it  affords  the  same  means  of 
escape  from  the  inevitable  distortions  of  plane  perspec- 
tive that  the  painter  of  figures  finds  in  the  corrections 
described  in  the  previous  chapter. 

285.  The  application  of  Plane  and  of  Cylindrical  Per- 
Rectif  in  spective  to  the  same  object  giving  such  differ- 
sketches.  ^^^^  results,  and  Cylindrical  Perspective  being, 
as  we  have  seen  (277),  the  method  one  naturally  adopts 
in  sketching  from  nature,  it  follows  that  any  attempt  to 
apply  the  principles  of  Plane  Perspective  to  sucli  sketches 
must  lead  to  confusion.  This  attempt  is,  however,  con- 
stantly made :  artists,  in  trying  to  avail  themselves  of 
the  materials  they  liave  collected  in  their  note-books, 
frequently  resorting  to  the  rules  of  perspective  to  cor- 
rect the  inconsistencies  and  errors,  and  to  fill  in  the 
omissions  of  the  originals.  The  vexation  and  trouble 
into  which  this  inevitably  brings  them  —  for  a  drawing 
made  upon  a  cylinder  cannot  be  treated  as  if  it  were 
made  upon  a  plane  —  has  produced  a  very  general  im- 
pression among  them  that  the  principles  of  perspective, 
though  true  in  theory,  as  they  say,  are  practically  false ; 
that  tliey  doubtless  work  very  well  for  geometrical  work, 
like  architectural  drawings,  but  that  when  applied  to 
nature,  to  the  delineation  of  real  things,  even  to  real 
buildings,  they  break  down. 

286.  It  is  indeed  plain  that  such  sketches  must  be 
interpreted  in  the  light  of  the  system  according  to  which 
they  were  made.     It  is  the  principles  and  rules  of  Cur- 


CYLINDiaCAL   PERSPECTIVE.  185 

vilinear  or  Panoramic  Perspective  that  must  be  called 
in.  This  being  so,  it  is  worth  while  to  inquire  on  what 
geometrical  principles  this  system  rests,  what  are  its 
practical  methods,  and  what  are  its  relations  to  the 
system  of  Plane  Perspective. 

287.  If  the  picture  is  supposed  to  be  drawn  not  upon 
a  vertical  plane,  but  upon  a  vertical  cylinder,  (jeo„,etricai 
with  the  spectator  at  the  centre,  or  axis,  it  is  p"''"p^«'- 
plain  that,  so  far  as  concerns  the  horizontal  dimensions 
of  objects,  their  perspective  representations  will  be  ex- 
actly proportionate  to  their  apparent  angular  dimen- 
sions, a  given  linear  measure  will  correspond  to  a  degree 
of  arc  on  the  horizon,  and  a  given  length  of  horizon  in 
the  picture  to  the  whole  circumference  of  360°.  Every 
part  of  this  horizon  will  be  equally  near  the  station- 
point,  and  every  point  in  it  may^  in  turn,  be  consid- 
ered as  'the  Centre.  We  thus  entirely  avoid  those 
distortions  which,  in  Plane  Perspective,  necessarily  re- 
sult from  the  visual  rays  crossing  the  plane  of  the 
picture  at  an  acute  angle,  as  they  must  do  everywhere 
except  just  opposite  the  station-point.  All  the  rays, 
at  least  the  rays  coming  from  all  points  of  the  horizon, 
cross  the  picture  at  right  angles ;  everything  is,  so  to 
speak,  at  the  Centre. 

And  just  as  in  Plane  Perspective  the  perspective  of  a 
straight  line  is  the  line  in  which  the  plane  of  the  pict- 
ure is  intersected  by  a  plane  of  rays  passing  through  the 
line  and  also  through  the  eye,  or  station-point ;  and 
just  as  the  horizon  of  every  system  of  planes  is  the  inter- 
section of  the  plane  of  the  picture  by  a  plane  passing 


18b'  MODERN   PERSPECTIVE. 

through  the  eye  parallel  to  the  other  planes  of  the  sys- 
tem ;  so  now  the  perspectives  of  lines,  and  the  horizons  of 
systems  of  planes,  are  the  lines  in  which  these  planes 
of  rays  intersect  the  cylinder  on  which  the  picture  is  to 
be  drawn.  This,  which  has  already  been  illustrated  in 
Plate  XVI  -^^8^'  ^^  ^^^^  ^^'  Plate  XV.,  is  more  fully  set 
^^s  ^^-  '  forth  in  Fig.  71,  Plate  XVI.  Here  S,  in  plan 
and  elevation,  represents  the  station-point  in  the  axis 
of  the  cylinder  of  the  picture.  If,  now,  a,  h,  c,  and  d, 
in  the  elevation,  are  parallel  horizontal  lines,  shown  in 
plan  at  L,  the  plane  of  rays  lying  between  them  and 
the  eye  will  cut  the  cylinder  A  at  a',  h',  c',  and  d',  the 
line  of  intersection  being  an  ellipse,  or,  at  c',  a  circle. 
If  the  cylinder  is  turned  a  quarter  round,  as  at  B,  these 
lines  of  intersection  will  appear  as  semi-ellipses  at  h" 
andc^";  the  semi-circle  c',  however,  appears  still  as  a 
straight  line  at  c",  while  the  line  a',  lying  at  an  angle  of 
45°,  appears  as  a  semi-circle  at  a". 

In  like  manner  the  line  e,  at  right  angles  to  these 
lines,  and  shown  in  plan  at  K,  gives  the  semi-circle  e"  in 
A  and  the  line  e'  in  B. 

288.  If  now  the  cylinder  is  developed,  these  lines  of 
The  develop-   intersection  will  appear  as  in  Pig.  72.  The  circle 

mentoftlie  ^^^    t  •if 

cylinder.  of  tlic  horizou  at  c  Will  l>ecome  a  straight  line ; 
the  vanishing-points  v^  and  v^,  v^'  and  v^',  will  appear  at 
V^  and  V^,  V^'  and  V^',  90°  apart;  while  the  semi- 
ellipses  a"  h"  d"  c"  will  be  developed  as  the  curves 
a'"  h'"  d'"  c'",  parallel  to  the  horizon   at  their 

Fig.  72.  '    ^ 

highest  point,  representing  the   point  where 
ahd  and  e  are  nearest  the  eye,  and  converging  towards 


CYLINDRICAL   PERSPECTIVE.  187 

their  vanishing-points,  as  parallel  lines  ought  to  do. 
The  height  of  the  lines  a'"  and  c!'\  representing  a  and  e, 
which  lay  45°  above  the  horizon,  is  equal  to  the  radius 
of  the  cylinder.  If  the  secant  planes,  a'  h' d\  Fig.  71,  had 
been  carried  entirely  across  the  cylinder,  forming  whole 
ellipses,  as  shown  by  the  dotted  lines,  these  curves 
would  be  continued  beyond  the  point  V^'  on  the  oppo- 
site side  of  the  horizon,  as  far  as  the  point  V^",  which 
is,  of  course,  the  same  point  as  V^  being  360°  distant 
from  it. 

289.  These  curves,  always  concave  towards  the  hori- 
zon, with   points    of  contrary  flexure   where 

Sine-curres. 

they  cross  it,  and  points  of  maximum  curva- 
ture where  they  are  at  their  greatest  distance  from  it, 
are  what  are  called  in  geometry  sine-curves,  or  cosine- 
curves,  and  are  similar  to  those  obtained  by  the  projec- 
tion of  a  regular  spiral.  It  will  be  observed  that  the 
angles  at  which  they  cross  the  horizon  are  the  same  as 
those  in  Fig.  71,  A,  and  measure  the  angular  distance  of 
the  lines  represented  above  the  horizon  at  their  nearest 
point.  The  maximum  distance  of  the  perspective  of 
each  line  from  the  horizon  is  proportional  to  the  tan- 
gent of  this  angle. 

290.  The  perspectives  of  vertical  lines  and  the  traces 
of  vertical  planes  are  of  course  vertical,  and  yj^rticai  and 
these  lines  are  straight,  being  elements  of  ]Tnes°and 
the  cylinder,  and  they  remain  straight  when  ^^^ ' 
the  cylinder  is  developed.  The  perspectives  of  inclined 
lines  and  the  traces  of  inclined  planes  are  sine-curves, 
since  they  too  are  the  lines  in  which  the  cylinder  is  cut 


188  MODERN   PERSPECTIVE. 

by  planes ;  but  tlieir  vanishing-points,  instead  of  being 
upon  the  horizon,  are  above  and  below  it,  as,  for  ex- 
ample, Y^  and  Y^'  in  Fig.  72.  But  they  are  all  parallel 
to  the  picture  at  the  point  where  they  are  nearest  to 
it  and  to  the  eye,  half  way  between  their  vanishing- 
points,  and  there  have  their  real  inclination. 

291.  It  is  a  property  of  a  sijdcm  of  sine-curves, — 
s  stems  of  ^^^^^  ^^  ^^  ^'"^^ '  ^  scrics  of  curvcs  crossing  the 
sine-curves.  j^Qj^j^on  at  the  samc  point,  like  the  perspec- 
tives of  a  system  of  parallel  lines,  —  that  vertical  lines 
drawn  across  them  are  divided  proportionally  to  the 
maximum  heights  of  the  curves,  and  that  for  each  ver- 
tical line  the  tangents  to  the  curves,  at  the  points  of 
intersection,  meet  at  the  same  point  on  the  horizon.  It 
is  necessary,  then,  to  construct  only  one  line  of  such  a 
system  by  developing  the  line  of  intersection.  The 
position  of  any  other  line,  of  which  the  angular  dis- 
tance above  the  horizon,  and  consequently  the  maxi- 
mum height,  is  known,  may  be  found  at  any  point  by 
To  construct  drawlug  a  vertical  line,  and  its  direction  at  that 

the  sine- 
curves,  point  may  be  found  by  drawing  a  tangent.     In 

tliis  respect  a  number  of  sine-curves  drawn  between  the 
same  two  points  are  analogous  to  a  number  of  ellipses 
with  the  same  major  axis  (265),  as  illustrated  in  Fig.  48, 
Plate  X. 

In  Fig.  73,  for  example,  tlie  distances  cut  off  by  the 
curves  upon  li  h  are  proportional  to  those  cut 
off  upon  1 1,  as  may  be  seen  at  t'  t',  and  the  tan- 
gents at  the  points  of  intersection  all  meet  at  Vp.     These 
tangent  lines  lie  in  a  plane  tangent  to  the  cylinder,  before 


CYLINDRICAL   PERSPECTIVE.  189 

it  is  rolled  out  flat,  along  the  line  1 1  This  relation  is 
shown  in  Fig.  71,  Plan,  from  which  it  is  clear  that  these 
tangent  lines  are  the  perspectives  upon  the  plane  ^9^9, 
regarded  as  a  plane  of  the  picture  tangent  to  the  cylin- 
der at  the  point  t,  of  the  same  lines  w^hose  perspectives 
are  given  upon  the  cylinder  by  the  curves.  Of  course 
they  meet  at  a  point.  This  point,  Y^,  is  then  the  van- 
ishing-point in  Plane  Perspective  of  the  same  lines 
whose  vanishing-point  is  found  in  Curvilinear  Perspec- 
tive at  V^.  The  plan  in  Fig.  66,  Plate  XV.,  exhibits 
similar  relations. 

292.  These  considerations  make  it  practicable  not 
only  to  draw  all  these  curves  without  difficulty  piane  Per- 
when  one  of  them  has  once  been  constructed,  anuxuiary 

1  •  •  T  •    1         n     T  method. 

but  m  practice  to  dispense  with  all  but  one   one  sine- 
curve  suffi- 

entirely,  treating  each  separate  portion  of  a  ^'®°*- 
panoramic  picture  as  if  it   were  drawn  on  a  vertical 
plane  tangent  to  the  cylinder  in  that  place. 

This  is  illustrated  in  Fig.  74,  a,  where  w^e  see,  as  in 
Fig.  66,  an  object,  L  E,  the  station  point  at  S, 

Fig.  74. 

and,  betw^een  them,  both  a  cylinder  and  a  plane 
of  projection.  The  object  not  being  very  large,  its  pic- 
ture on  tlie  cylinder  will  not  differ  perceptibly  from  its 
picture  upon  the  plane ;  and  the  latter,  being  easier  to 
draw%  may  be  substituted  for  the  other,  using  Vp  for  Vq, 
and  V?  for  Vg. 

293.  Fig.  74,  h,  shows  that,  in  laying  out  a  panorama, 
it  is  not  necessary  to  draw  even  a  single  sine-  Nosine-curre 
curve,  the  points  Yp  and  Yp  being  ascertained  '^^^^^^^^' 


190  MODERN   PERSPECTIVE. 

directly  from  VJ  and  Vf,  in  accordance  with  the  fol- 
lowing rule :  — 

Given  the  Horizon,  the  vertical  line  of  tangency,  and 
To  find  the  ^^^®  vanishing-points  on  the  cylinder,  set  off 
vanishTag.  bclow  thc  Horizon,  on  the  vertical  line  of  tan- 
points.  gency,  the  radius  of  the  cylinder;  with  this 

radius  describe  an  arc  tangent  to  the  Horizon ;  on  this 
arc  lay  off,  right  and  left,  the  distances  of  Vg  and  V^; 
prolong  the  radii  through  the  extreme  points  thus  at- 
tained, and  the  points  wliere  they  strike  the  Horizon 
will  be  the  vanishing-points  required,  VP  and  Vp,  on 
the  Horizon  of  the  tangent  plane.  The  rest  of  the  con- 
struction can  then  take  place  as  in  Plane  Perspective. 

This  rule  explains  itself  if  we  observe  that  Fig.  74,  h, 
is  the  same  as  Fig.  74,  a,  with  the  two  parts  brought 
together. 


'O^ 


294.  Fig.  75,  which  is  taken  from  the   etching  of 

Greenw^ich  Hospital,  in  Turner's  "  Liber  Stu- 

Fig.  75.          .  ^  . 

diorum,"  further  illustrates  the  occurrence  of 
curved  lines  in  sketching  from  nature,  when  every 
object  is  drawn  of  just  the  shape  and  relative  size  that 
it  presents  to  the  eye;  that  is,  in  proportion  to  its 
angular  dimensions. 

295.  In  a  drawing  upon  a  vertical  cylinder,  although 

the  horizontal  dimensions  are  proportional  to 

The  use  of  a  ^  x       ir 

c^iinder^^      the  augular  dimensions  of  the  objects  repre- 
sented,  vertical   dimensions  are  proportional 
to  the  tangents,  as  upon  a  plane.     If  it  were  practicable 


CYLINDRICAL   PERSPECTIVE.  191 

to  draw  upon  a  spherical  surfece,  this  would  not  be  so, 
the  linear  dimensions  in  every  direction  being  made 
proportional  to  the  angular  dimensions.  Tlie  employ- 
ment of  a  horizontal  cylinder  to  draw  upon,  or  in,  would 
of  course  effect  this  for  vertical  dimensions.  Tliis  is 
practicable' for  isolated  objects,  and  the  result  may  some- 
times be  seen  in  sketches  of  lofty  buildings  or  towers. 
The  curvature  of  the  lines,  and  the  apparent  convexity 
of  the  object,  making  the  tower  look  as  if  it  were  strut- 
ting or  leaning  over  backwards,  are,  however,  quite  as 
offensive  as  in  the  case  of  horizontal  lines.  But  here,  as 
in  the  case  of  the  street  shown  in  Fii^.  G9,  these  defects 
disappear  when  the  object  is  of  irregular  outline ;  and 
this  method  has  the  advantage,  of  course,  of  giving  the 
actual  aspect  of  a  tower  or  spire  from  a  given 
point,  with  its  foreshortening  and  altered  out- 
line, better  than  the  other.  Fig.  76  illustrates  this,  show- 
ing at  A  a  view  of  the  spire  of  the  Central  Church  in 
Boston,  taken  from  a  photograph,  and  consequently 
drawn  on  the  principles  of  Plane  Perspective.  At  B 
is  a  sketch  of  the  same  spire,  taken  at  a  correspond- 
ing distance,  drawn  upon  the  surface  of  a  horizontal 
cylinder  with  a  revolving  camera  obscura. 


CHAPTEE   XTV. 

DIVERGENT   AND    CONVERGENT    LINES.  —  REFLECTIONS.  — 
SHADOWS    BY   ARTIFICIAL   LIGHT. 

THE  phenomena  with  which  perspective  has  to  do 
ave  mainly  the  phenomena  of  parallel  lines,  —  lines 
which  seem  to  meet  at  an  infinitely  distant,  or  vanishing- 
point  ;  and  we  have  seen  that  the  discussion  of  such  lines 
prepares  the  way  for  the  elucidation  of  the  rather  com- 
plicated phenomena  of  shadows,  since  the  rays  of  the 
sun,  and  the  shadows  caused  when  these  rays  are  inter- 
rupted, are  also  parallel  riglit  lines  belonging  to  a  sin- 
gle system.  But  it  is  worth  while  to  examine  also  the 
phenomena  of  divergent  or  convergent  lines,  —  of  lines 
tliat  is  to  say,  which  actually  do  meet,  or  tend  to  meet, 
at  a  point  which  is  within  a  finite  distance.  These 
phenomena  so  closely  simulate  the  phenomena  of  par- 
allel lines  that  we  may  well  inquire  just  how  far  the 
resemblance  extends.  Moreover,  just  as  the  sun  and 
moon  cast  parallel  shadows,  every  terrestrial  source  of 
light,  being  at  a  finite  distance,  casts  divergent  shadows, 
and  it  is  a  matter  of  practical  importance  to  find  out 
just  how  they  go,  and  what  becomes  of  them. 

296.     Now  just  as  parallel  lines  seem  to  converge, 
and  look   exactly  as  if  they  were  radiating  in   every 


DIVERGENT   AND    CONVERGENT   LINES.  193 

direction,  from  some  point  not  very  far  off,  situated  upon 
the  line  between  the  eye  and  the  infinitely  dis-  conyergent 
taut  vanishing-point,  —  so  lines  that  do  radiate  ^*"^^' 
irom  a  point  more  or  less  distant  look  exactly  as  if  they 
were  parallel  lines  going  oft'  to  infinity.  And  as  a  per- 
spective drawing  has  to  do  only  with  the  appearances 
of  things,  it  represents  the  two  classes  of  phenomena  in 
exactly  the  same  way.  All  it  can  do,  in  either  case,  is 
to  show  a  system  of  right  lines,  all  meeting, 

The  apex. 

or  tending  to  meet,  at  the  same  point.     The 

point  which  thus  simulates  a  vanishing-point  we  wdll 

call  the  a2:)ex  of  the  converging  lines. 

297.  This  consideration  leads  at  once  to  the  solu- 
tion of  a  question  in  itself  somewhat  puzzling.  If  the 
apex,  or  point  in  which  tlie  lines  meet,  is  in  front  of  the 
spectator,  it  is  easy  enough  to  draw  the  lines  of  the  con- 
vergent system  radiating  from  the  perspective  of  the 
apex,  like  the  spokes  of  a  wheel.     But  what  The  apex 

^        '  ^  _  behind  the 

shall  we  do  if  the  apex  is  behind  us,  the  lines  spectator. 
being  divergent,  passing,  so  to  speak,  over  our  head  and 
around  our  shoulders,  and,  as  they  recede  from  us  in 
front,  separating  from  each  other  more  and  more.  Such 
are  the  rays  emitted  by  a  candle  set  behind  one's  back, 
and  the  shadows  cast  by  it.  What  becomes  of  these 
divergent  beams  ?  Where  do  they  seem  to  go  ?  How 
shall  their  perspectives  be  drawn  —  so  much  of  them 
as  extends  beyond  the  plane  of  the  picture  —  upon  that 
plane  ? 

298.  The  observation  made  just  now  as  to  the  exact 
resemblance    between  the  phenomena  of   divergent   or 

13 


194  MODERN   PERSPECTIVE. 

convergent  lines,  and  the  phenomena  of  parallel  lines, 
affords  a  hint  of  the  answer  to  this  question.  Par- 
allel lines  have  two  vanishing-points,  180°  distant  one 
from  the  other,  upon  a  line  passing  through  the  station- 
point.  If  the  spectator,  looking  in  the  direction  of  the 
parallel  lines  of  the  system,  sees  one  of  these  vanishing- 
points  directly  before  him,  the  other  will  be  directly 
behind  him ;  and  on  turning  right-about-face  he  will 
see  that  instead.  So  with  the  rays  of  the  sun.  One 
vanishing-point  is  in  the  sun  itself;  the  other,  as  we 
have  seen,  is  in  exactly  the  opposite  direction,  namely, 
in  the  shadow  of  the  spectator's  head  (170). 

But  as  divergent  lines  exactly  simulate  these  phe- 
nomena, and  cannot  in  fact  be  distinguished  in  appear- 
ance from  parallel  lines,  except  by  some  extraneous 
indication,  they  too  will  seem  to  have  a  second  point  of 
convergence,  opposite  the  apex,  towards  which  they 
seem  to  tend ;  and  when  the  sj)ectator  turns  his  back 
upon  the  apex,  and  looks  in  exactly  the  opposite  direc- 

The  false  ^^^^^>  ^^^  ^^^^^  ^^^  ^^^®  positiou  of  this  falsc  apex 
*P^^-  directly  before  his  eyes.     So  with  rays  of  a 

candle,  or  any  source  of  artificial  light.  If  he  faces  it, 
the  rays  converge  upon  it.  If  he  turns  his  back  upon 
it,  they  seem  to  be  directed  towards  the  point  exactly 
opposite  the  candle.  This  point,  as  before,  is  to  be 
found  in  the  shadow  of  the  spectator's  head,  in  that 
part  of  the  shadow,  namely,  that  we  may  call  the 
shadow  of  his  eye.  That  lines  diverging  from  a  point 
behind  the  spectator  must  seem  to  converge  towards  a 
point  in  front  of  him  is  plain ;  for  if  we  suppose  a  line 


DIVERGENT   AND    CONVERGENT   LINES.  195 

to  pass  through  the  eye,  in  any  direction,  all  lines  lying 
in  planes  that  pass  through  this  line  will  seem  to 
be  directed  towards  one  or  the  other  of  its  vanish- 
ing-points. But  the  line  drawn  through  the  station- 
point  from  the  apex  of  these  divergent  lines  is  sucli 
a  line  passing  through  the  eye,  and  the  divergent 
lines  are  lines  lying  in  such  planes.  They  necessarily 
tend,  either  way,  towards  the  vanishing-points  of  the 
line.  Of  these  vanishing-points,  one  will  be  in  the 
direction  of  the  apex,  the  other  in  exactly  the  opposite 
direction.     This  will  be  the  false  apex. 

299.    Fig.  77,  Plate  XVII.,  illustrates  these  points,  and 
also  brings  into  notice  the  curious  phenomena  piatexvii 
that  present  themselves  when  the  apex,  or  point  ^'^'  '^' 
of  convergence,  is  neither  behind  the  spectator  nor  in 
front  of  him,  but,  so  to  speak,  along-side ;  that  is  to  say, 
is  just  as  far  from  the  plane  of  the  picture  as  The  apex  iu 

three  posi- 

he  is  himself.  Let  S  be  the  station-point,  p  p  tious. 
the  plane  of  the  picture,  and  1,  2,  3,  4,  5,  6,  7,  8,  9,  and 
10,  ten  right  lines  lying  in  a  horizontal  plane  somewhat 
below  the  level  of  the  eye,  one  end  of  each  touching  the 
line  ^^  in  plan,  and  the  ground-line,  g  I,  in  the  per- 
spective view  above.  A  simple  inspection  of  the  figure 
show^s  that  the  parallel  lines  1,  2,  and  5  have  their  van- 
ishing-point at  V^ ;  3,  4,  6,  8,  and  9,  which  are  perpen- 
dicular to  the  plane  of  the  picture,  have  their  vanishing- 
point  at  V^;  while  7  and  10  have  theirs  at  V^.  The  apex 

behind  the 

Moreover,  the  lines  1,  3,  and  7,  which  con-  spectotor. 
verge  upon  Aj,  behind  the    station-point,  and   which 
accordingly  appear  to  the  spectator  at  S  to  be  diver- 


196  MODERN   PERSPECTIVE. 

gent,  liave  their  perspectives  directed  towards  A;,  a 
false  apex,  a  point  in  space  exactly  opposite  the  apex 
itself,  and  as  far  above  the  horizon  as  the  apex  Aj,  on 
the  ground,  is  below  it. 

The  lines  5,  6,  and  7,  which  converge  upon  A2,  the 
The  a  ex  in  ^P^^  iu  fVout  of  tlic  spcctator,  are  shown  in 
^'"°°''"  perspective  meeting  at  the  perspective  of  Ag. 

300.  Finally,  the  lines  5,  8,  and  10,  which  have  their 
apex  at  A3,  a  point  just  as  far  from  the  plane  of  the  pic- 
T'leapex  ^^^^^  ^^  ^^^®  statiou-poiut,  have  their  perspec- 
aioug-side.  ^..^^^y  parallel.  The  same  is  true  of  tlie  lines  2, 
4,  and  7,  which  converge  upon  the  horizontal  projection 
of  the  station-point,  at  S.  But  these  last  lines  are  not 
only  j)arallel,  but  vertical. 

301.  It  appears,  then,  that  if  converging  lines  have 
their  apex,  or  point  of  meeting,  in  front  of  the  spectator, 
their  perspectives  will  converge  upon  the  perspective  of 
their  apex ;  if  it  is  behind  the  spectator,  they  will  con- 
verge upon  a  point  which  we  have  called  their  false 
apex,  situated  exactly  opposite  the  apex  itself,  and  which 
is  the  vanishing-point  of  a  line  drawn  through  the  apex 
and  the  station-point ;  if  the  apex  is  neither  in  front  of 
the  spectator  nor  behind  him,  but  just  as  far  from  the 
picture  as  is  the  station-point,  the  perspectives  of  the 
convergent  lines  will  be  parallel.  Their  position  de- 
pends upon  the  height  at  which  the  apex  lies,  and 
is  most  easily  determined  by  finding  the  perspective  of 
that  element  of  the  convergent  system  that  is  perpen- 
dicular to  the  plane  of  the  picture  (such  as  line  8  in  the 
figure),  and  making  the  rest  parallel  to  that.     Finally, 


REFLECTIONS.  197 

if  the  apex  is  directly  above  or  below  the  station- 
point,  the  perspectives  of  the  convergent  lines  will  be 
vertical.  If  it  is  on  a  level  with  his  eye,  they  will  be 
horizontal. 

302.   Fig.  78,  a  sketch,  somewhat  reduced  in  scale, 
from  an  old  English  print,  illustrates  this  last 

-Til  •  *'jg-  '8. 

pomt.    it  shows  three  streets  converging  upon 

the  point  occupied  by  the  spectator.     The  axes  of  these 

streets  are  all  drawn  vertical  and  parallel. 

A  similar  result  would  follow  if  one  were  to  draw  the 
spokes  of  a  large  wheel  while  standing  upon  the  hub. 
They  would  all  be  vertical  and  parallel.  So  with  the 
reflections  of  distant  lamps  which  at  night  often  seem 
to  cross  the  water  and  converge  below  the  feet  of  a 
spectator  standing  upon  a  bridge.  In  perspec-  Reflections, 
tive  they  also  would  all  be  drawn  vertical  and  Fig-  'i^- 
parallel,  continuing,  indeed,  the  vertical  line  of  the  lamp- 
posts from  which  they  emanate,  as  is  seen  in  Fig.  79. 

303.    Fig.   80  shows  that  the  position  of  the  false 
apex   may  most  conveniently  be  determined  to  find  the 
by  finding  the  perspectives  oi  two  horizontal  perspective. 
elements  of  the  converiiinGj   series,  a  and   h,     ^Js  ^o. 
one  perpendicular  to  the  plane  of  the  picture,  and  one 
passing  beneath  (or  above)  the  station-point.       Let  a' 
and  y  (or  a"  and  h")  represent  the  points  in  which  these 
lines  pierce  the  plane  of  the  picture,  their  distance  from 
the  horizon  depending  iij)on  the  position  of  the  apex,  A, 
below  or  above  the  eye,  and  the  other  element  of  their 
position   being   obtained   from  the   orthographic   plan. 


198  MODERN   PERSPECTIVE. 

The  perspective  of  the  line  a  will  begin  at  a'  (or  a"), 
and  be  directed  towards  its  vanishing-point,  Y^  ;  that  of 
the  line  6  will  begin  at  V  (or  5"),  and  be  vertical,  since  it 
passes  directly  below  (or  above)  the  station-point  (301). 
The  point  A'  (or  A"),  where  these  perspectives  would 
meet,  if  prolonged,  is  the  position  of  the  false  apex. 

The  same  result  may  be  obtained  by  the  methods  of 
Another  way,  Descriptivc  Gcomctry,  as  may  be  seen  in  the 
geometry.  sauic  hgurc.  The  false  apex  is  the  vanishing- 
point,  as  has  just  been  seen  (298),  of  that  element  of  the 
converging  system  that  passes  through  the  eye.  But 
the  vanishing-puint  of  a  line  passing  through  the  eye, 
and  which  is  consequently  seen  endwise,  is  the  point 
where  it  pierces  the  plane  of  the  picture.  Now  5,  in 
Fig.  80,  is  the  horizontal  projection  of  such  a  line,  and 
a'  A'  (or  a"  A")  its  vertical  projection.  A'  then  (or  A") 
is  the  point  where  it  pierces  the  picture,  its  vanishing- 
point,  and  the  false  apex  in  question. 

304.    Fig.  77  furnishes  other  illustrations  of  the  false 

apex  besides  those   already  mentioned.     The 

points  of  convergence,  A5,  Ag,  and  A7,  situated 

behind  the  spectator,  have  each  a  false  apex,  towards 

which   the   perspectives    of    the   converging   lines   are 

directed,  as  may  be  seen  at  A5',  Ag',  and  A7'. 

A5,  Ag,  and  A7  are  supposed  to  lie  in  the  same  plane 
as  do  Ai,  A2,  A3,  and  A4,  previously  discussed.  It  will 
be  observed  that  the  most  distant  of  them,  Ag  and  A7, 
have  the  apex  nearest  the  horizon,  the  nearer  ones,  Aj 
and  A5,  farther  off,  and  tliose  on  a  line  w^ith  the  spec- 


SHADOWS    BY   ARTIFICIAL   LIGHT.  199 

tator,  A3  and  A4,  at  an  infinite  distance,  the  perspectives 
of  the  conv^erging  lines  being  parallel. 

If  either  apex  were  in  a  different  horizontal  plane, 
the  vertical  position  of  the  false  apex  would  change 
accordingly,  that  of  A*^,  for  instance,  appearing  at  A,"  if 
the  point  of  convergence  were  as  far  above  the  specta- 
tor's level  as  it  has  been  supposed  to  be  below  it. 

305.  In  determining  the  form  of  sliadows  cast  by 
artificial  light  we  follow  the  same  line  of  argu-  shadows  by 

artificial 

ment  as  when,  in  Chapter  IX.,  we  discussed  "giit. 
the  shadows  cast  by  the  sun.  Here,  as  there,  the 
(invisible)  shadow  of  a  point  is  a  line  drawn  through 
the  point  from  the  source  of  light,  and  its  (visible) 
shadow  on  any  surface  is  the  point  where  this  line 
pierces  the  surface.  Here,  as  there,  the  (invisible) 
shadow  of  a  line  is  a  plane  in  space,  and  its  (visible) 
shadow  upon  any  plane  that  receives  it  is  the  line  of 
intersection  of  these  two  planes,  its  vanishing  point 
being  at  the  intersection  of  their  traces.  Here,  as  there, 
the  horizon  of  the  plane  of  shadow  is  found  by  drawing  a 
line  tlirough  the  vanishing  points  of  two  of  its  elements, 
one  being  the  vanishing  point  of  the  line  that  casts 
the  shadow,  and  the  other  the  vanishing  point  of  the 
(invisible)  shadow  of  some  point  in  that  line. 

306.  The  only  difference  in  the  two  cases  is  that  in 
the  case  of  sunlight  the  vanishing  point  of  this  last  line 
is  always  known  beforehand.  The  shadows  of  all  the 
points  are  parallel  lines,  and  have  a  common  vanishing 
point,  either  in  the  sun,  or  in  the  shadow  of  the  specta- 


200  MODERN   PERSPECTIVE. 

tor's  head,  opposite  the  sun.  But  with  artificial  light 
the  lines  drawn  from  the  source  of  light  to  the  different 
points  of  the  line  whose  shadow  is  to  be  found  are  not 
parallel  but  divergent.  Each  has  a  direction  of  its  own 
and  a  vanishing  point  of  its  own.  Still,  all  these  di- 
vergent lines  lie  in  the  same  plane,  the  plane  of  shadow 
in  question,  like  the  sticks  of  a  fan,  and  any  one  of  them 
will  suffice,  with  the  line  itself,  to  determine  the  horizon 
of  this  plane.  For  this  it  is  necessary  only  to  select  some 
one  of  these  divergent  lines  and  determine  its  vanishing 
point. 

307.  This  does  not  of  course  lie  in  the  apex  of 
To  find  the  these  converging  rays,  the  source  of  light  itself, 
pdintofa'ray  It  uiust  bc  determined  just  as  the  vanishing 

of  artificial  .  ,.,.., 

light.  point  of  any  finite  line  is  determined,  when 

Fig.  81,  a.  |-|-^g  |-^-^Q  -g  gi^Q^^  \)y  l^^Q  points  at  its  extremi- 
ties. *In  Fvy  81,  a,  for  instance,  let  M  be  the  line 
whose  shadow  is  to  be  cast,  determined  in  position  by 
the  vertical  lines  at  its  extremities,  which  show  its  rela- 
tion to  the  horizontal  plane.  Let  A  be  the  apex  of  rays, 
or  source  of  light,  and  let  a  single  ray,  S',  be  drawn 
through  this  point  and  some  point  upon  the  given  line, 
say  its  lowest  point.  These  two  lines  suffice  to  deter- 
mine tlie  plane  of  shadow,  and  tlie  horizon  of  tliat  plane, 
H  S  M,  will  pass  througli  their  vanisliing  points,  Y^' 
and  V^. 

308.  Fig.  81,  Z),  shows  how  these  may  be  determined. 

The  horizontal  line  E,  connectinoj  the  perpen- 

Fig.  81,  fc.  o  r      r 

diculars  let  fall  from  the  ends  of  the  line  M, 
being  prolonged  until  it  touches  the  horizon,  finds  its 


SHADOWS    BY    ARTIFICIAL    LIGHT.  201 

vanishing-point  at  V^  (for  a  line  lying  in  a  plane 
always  has  its  vanishing-point  in  the  horizon  of  that 
plane) ;  the  horizon  of  the  vertical  plane  that  contains 
M  and  R  passes  vertically  through  Y^  (for  the  ^^  ^^^^  ^^^^ 
horizon  of  a  plane  passes  through  the  vanish-  SpiTufof 
ing-points  of  every  line  that  lies  in  it) ;  and 
the  line  M  prolonged  finds  its  vanishing-point,  V^,  on 
this  horizon  (the  horizon  of  the  vertical  plane  in  which 
it  lies). 

In  the  same  manner  Y^'  is  found  by  drawing,  as  far 
as  the  horizon,  a  line  beneath  the  ray  of  light,  and  by 
producing  that  ray,  until  it  meets  the  hoiizon  of  the 
vertical  plane  in  which  it  lies.  H  S  M,  tlie  horizon  of 
the  shadow  of  M,  is  then  found,  as  usual,  by  drawing  a 
line  between  Y^'  and  Y^. 

The  visible  shadow  of  the  line  M  upon  the  plane, 
S  M  on  R  L,  lies  at  the  intersection  of  the  plane  of 
invisible  shadow  with  the  ground  plane.  Its  vanishing- 
point,  Y^^,  ^^  is  the  intersection  of  their  horizons  ;  that 
is  to  say,  of  the  horizons  H  S  M  and  H  R  L.  The  visible 
shadow  will  begin  wdiere  the  ray  S',  passing  through  the 
end  of  the  line  M,  reaches  the  ground,  and  will  be 
directed  towards,  or  away  from,  this  vanishing-point,  as 
in  the  figure. 

A  similar  result  would  have  been  reached,  as  appears 
in  the  figure,  by  taking  rays  of  light  that  pass 
through  other  points  of  the  line  M,  such  as  S'^  or  S"'. 
These  have  different  directions,  but  lie  in  the  same 
plane  ;  this  gives  us  new  vanishing-points,  Y^"  and  Y^"', 
but  the  same  horizon,  H  S  M,  just  as  before. 


202  MODERN    PERSPECTIVE. 

309.    Any  other  ray  would  answer  as  well,  of  course, 
as  these  ;  the  choice  of  one  or  another  is  purely 

A  second  way;  '  ^  "^ 

leUoYhrpk-  a  matter  of  convenience.  Now  just  as  we 
*'"'^'  found,  in  studying  shadows  cast  by  the  sun, 

that  they  were  most  easily  determined  when  the  rays 
of  light  were  parallel  with  the  picture  (the  sun  being 
neither  behind  the  spectator  nor  in  front  of  him,  but  on 
one  side),  so  here  we  shall  find  that  there  is  a  practical 
advantage  in  selecting  for  use  neither  of  the  rays  just 
now  employed,  but,  instead,  the  ray  that  is  parallel  to 
the  picture,  the  ray  that  passes  through  the  line  at  a 
point  which  is  just  as  far  from  the  picture  plane  as  is 
the  apex,  A,  or  source  of  light. 

Fig.  81,  c,  shows  how  this  is  done.     A  line  parallel 
to  the  horizon  is  drawn  through  the  foot  of 

Fig.  81,  c. 

the  vertical  let  fall  from  the  apex  A  until  it 
intersects  the  line  E,  at  wliich  point  another  vertical  is 
erected  to  intercept  tlie  line  M.  The  line  S,  from  A  to 
this  point,  is  obviously  a  ray  of  light  parallel  to  the 
picture,  and  the  line  drawn  through  Y^,  parallel  to  it, 
is  tlie  horizon  of  the  plane  of  shadows,  H  S  M. 

This  is  exactly  the  method  employed  in  Fig.  36, 
Plate  VIIL,  founded  on  the  proposition  that  the  ele- 
ment of  a  plane  parallel  to  the  picture  has  its  perspec- 
tive parallel  to  the  horizon  of  the  plane  (38). 


310.    The  relations  between  the  treatment  of  divers- 
ing   light   and  that  of  parallel  liijht  mav  be 

Fig.  81,  rf.     .  ^  o  J 

illustrated  by  these  figures.     If  the  source  of 
light.  Fig.  81  d,  be  supposed  to  retreat  to  an  infinite 


SHADOWS    BY    SUNLIGHT.  203 

distance,  the  vertical  line  dropped  upon  the  horizontal 
plane  will  shorten  until  it  rests  upon  the  horizon. 
ys/^  ys//^  g^j-j^  V^"',  will  coalesce  with  each  other  and 
with  A,  which  will  now  become  Y®,  and  the  figures  will 
both  assume  the  aspect  of  Fig.  81,  c?. 

311.  In  casting  shadows  by  artificial  light,  however 
it  is  not  so  important  to  obtain  the  vanishing-point  of 
the  line  of  visible  shadow  as  it  is  in  sunlight.  For,  in 
sunlight,  the  vanishing-point  of  the  shadow  of  a  line, 
when  once  obtained,  serves  for  the  shadows  of  all  the 
other  lines  of  the  system,  since  the  shadows  are  parallel 
as  well  as  the  lines  themselves,  and  all  have  the  same 
vanishing-point.  But  in  artificial  light  the  shadows  of 
parallel  lines  are  not  parallel,  and  the  vanishing-point 
of  one  is  of  no  service  in  drawing  the  next  one. 

It  is  accordingly  just  as  well,  and  it  is  much  easier, 
to  find  the  shadow  cast  upon  a  plane  by  a  line 

i-  ^  '^  Fig.  81,  «. 

exposed  to  artificial  light  by  finding  the  shad- 
ows of  its  extreme  points  and  drawing  a  staight  line 
between  them.  Fig.  81  c  shows  how  this  may  be  done. 
The  point  in  wliich  the  ray  S'  touches  the  ground  is  one 
point  of  the  line  of  shadow,  the  ray  S^'  gives  a  second 
point,  and  the  point  where  the  line  itself  pierces  the 
ground  gives  a  third.  Either  two  of  these  suffice  to  fix 
the  line  of  shadow  in  position  and  direction. 

Fig-.  81  /  shows  that  in  the  sunlight,  also,  shadows  by 

Sunlight. 

we  can  in  like  manner  determine  a  shadow  by  Fig.  si,/ 
points,  without  using  the  vanishing-point. 

This  is,  whether  in  sunlight  or  artificial  light,  the 
most  convenient  way  of  determining  the  shadow  of  a 


204  MODERN    PERSPECTIVE. 


curv^ed  or  irregular  line.  It  amounts,  virtually,  as  tlie 
fio'ure  shows,  to  lettino-  fall  a  i)erpeiidicular 
iroin  eacJi  point  upon  the  plane  in  question, 

and    drawing   its  shadow,  and  then  passing  a   line  of 

shadow  through  the  extreme  points  thus  obtained,  as  is 

done  in  Fig.  81  ^. 


312.    But  a  plane  given  by  a  line  and  a  point  can 

Fig  81, -/i.    be  determined  by  a  line  drawn  through   the 

the  ray  pari  poiiit  parallel  to  the  oiven  line,  as  well  as  by 

allel    to   the    ^  ^  ^  *^ 

given  line.  oue  crossiiig  it.  The  same  result  as  in  the 
previous  figures  may  accordingly  be  obtained  by  em- 
ploying the  method  shown  in  Fig.  81,  h. 

The  point  V^  having  been  determined,  a  ray,  S^,  par- 
allel to  M,  is  drawn  tlirough  the  apex  A,  touching  the 
ground  at  its  point  of  intersection  with  the  horizontal 
line  beneath  it,  the  line  M  being  also  prolonged  to  a 
similar  point.  This  ray,  the  line  M,  and  the  horizontal 
line,  L,  joining  their  extreme  lower  points,  are  obviously 
all  in  the  plane  of  shadow. 

The  line  prolonged  in  the  other  direction,  beyond  M, 
beinc:  the  line  in  which  the  shadow  of  M  intersects  the 
horizontal  plane,  is  obviously  the  visible  shadow  of  M 
on  that  plane,  the  exact  length  of  which  can  be  cut  off 
by  drawing  divergent  rays  from  A  through  the  extremi- 
ties of  M,  as  in  the  figure. 

The  shadow  of  any  line  cast  on  any  plane  surface  by 
an  artificial  source  of  light  may  accordingly  be  found 
by  drawing  a  single  ray  parallel  to  the  line  in  question, 
and  finding  the  points  in  which  both  the  line  in  question 


SHADOWS   BY    ARTIFICIAL   LIGHT.  205 

and  the  ray  will  pierce  the  plane  on  xA'hich  the  shadow 
falls.     A  line  joining  these  points  will  be  the  ^g^^erai 
line  in  which  the  plane  of  rays  passing  througli  °*^*^"^- 
the  line  intersects  this  plane,  and,  if  produced  beyond 
the  point  where  the  line  pierces  the  plane,  will  give  the 
line  of  its  shadow. 

This  method  seems  less  direct  than  that  of  Fig.  81  e, 
which  is  indeed  often  preferable  for  single  lines.  But 
the  method  just  explained  is  to  be  preferred  where  sev- 
eral parallel  lines  are  subjected  to  artificial  light.  For 
the  point  x,  Fig.  SI  h,  where  the  parallel  ray  pierces 
the  plane  upon  which  the  shadow  falls,  holds  the  same 
relation  to  one  line  of  tlie  system  as  to  another,  the 
shadows  of  all  of  them  passing  through  it.  The  shadow 
of  the  line  M2,  parallel  to  M,  in  tlie  figure,  is  directed 
towards  this  point.  It  is  then  an  apex  of  converging 
lines,  and  is  as  important  and  serviceable  a  point  when 
the  shadows  of  parallel  lines  are  cast  by  artificial  light, 
as  their  vanishing-point  is  when  they  are  cast  by  sun- 
light, and  answers  the  same  purpose  in  saving  labor. 

313.  It  a}jpears  then,  that,  in  artificial  lights  the 
shadows  upon  a  given  iilane  of  a  given  system  of  par- 
allel lines  all  converge  to  the  point  in  the  plane  at 
ichich  it  is  pierced  hy  an  element  of  tlie  system  pass/d 
through  the  source  of  light. 

314    Fig.  81  i,  shows  that  the  method  of    pig. sit. 
Fig.  81  h,  may  also  be  used  for  findino-  shadows  Thismethod 

^  '  «^  ^  applied  to 

by  sunlight,  the  shadow  of  a  line  in  a  plane  ^"°"g^*- 
being  determined   by  means  of  a  second  line  parallel 
with  it,  instead  of  by  a  ray  of  light  crossing  it. 


206.  MODEKN   PERSPECTIVE. 

We  cannot,  to  be  sure,  draw  this  second  line  through 
the  source  of  light,  that  being  infinitely  distant.  But 
it  is  not  essential,  even  with  artificial  light,  that  the 
parallel  line  be  drawn  througli  the  apex  of  the  diver- 
gent rays.  It  may  be  drawn  through  any  point  of  any 
ray.  In  Fig.  81,/,  for  instance,  a  point  being  taken  on 
a  ray  that  passes  from  A  to  the  upper  end  of  M,  its 
height  above  the  plane  is  easily  determined,  and  a  line 
parallel  to  M  can  as  serviceably  be  talvcn  through  that 
point  as  through  A  itself. 

But  this  process  can  be  exactly  as  well  employed  with 
the  parallel  rays  of  the  sun,  as  appears  in  Fig.  81,  ^, 
where  the  shadow  of  M  on  the  ground  is  again  found 
without  the  aid  of  the  horizon  H  S  M. 

315.  Fig.  82,  Plate  XVIIL,  iUustrates  the  ease  with 
wdiich  can  be  found  the  shadows  of  vertical  lines  upon 
a  horizontal  plane.  The  sliadow  of  each  such  line  is 
obviously  a  vertical  plane,  of  which  the  vertical  line 
Plate xvni  clropped  from  the  source  of  light.  A,  is  also 
Thesiiatiow    au  clcmeut.      The  line  joining  the  lower  ends 

of  vertical 

lines.  of  these  vertical  lines  is  accordingly  the  in- 

^'^"  ■  tersection  of  the  plane  of  shadow  with  the 
ground,  and  its  prolongation  gives  the  line  of  shadow. 
This  has  indeed  been  done  incidentally,  as  was  said,  in 
Figs.  81,  e  and  //. 

Tlie  figure  shows,  also,  that  a  similar  procedure  may 
be  followed  in  the  case  of  a  liorizontal  line,  normal  to  a 
vertical  plane,  the  projection  of  the  apex  of  rays  upon 
that  plane  being  known.     (313.) 


SHADOWS   BY   ARTIFICIAL   LIGHT.  .207 

316.  The  principle  of  Fig.  81  h  is  illustrated  in  Fig. 
83,  where  the  shadows  of  several  parallel  lines  „,  ^ 

'  i  Shadows  of 

are  thrown  upon  a  broken  surface  composed  p^^'^^'^i  ^^^^s. 
of  several  planes.  The  shadow  of  each  line  is  '^' 
directed,  in  each  plane,  towards,  or  away  from,  the  point 
where  a  ray  drawn  parallel  to  it  through  the  apex  of 
rays,  or  source  of  light,  pierces  that  plane.  As  the  same 
ray  is  parallel,  of  course,  to  all  tlie  lines  of  the  system, 
one  ray  suffices  for  them  all,  and  their  shadows,  in  each 
plane,  are  directed  towards  the  same  point. 

The  vertical  and  inclined  lines  that  cast  the  shadow 
lie  in  a  vertical  plane,  and  the  vertical  and  inclined  rays 
taken  parallel  to  them  lie  in  a  vertical  plane  parallel  to 
it.  The  points  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11,  in  which  these 
rays  pierce  the  planes  on  which  the  shadows  fall,  lie  in 
the  line  in  which  the  plane  of  rays  cuts  those  planes. 

Of  these  points,  1,  2,  and  3  are  those  in  which 
the  vertical  ray  pierces  the  ground  and  the  two  in- 
clined planes  at  the  bottom  of  the  wall,  and  the  shad- 
ows of  the  vertical  bars  cast  upon  these  planes  are 
directed  to  these  points ;  4,  5,  and  6  are  tlie  points  at 
which  the  horizontal  ray  pierces  the  three  vertical  planes 
of  the  wall  and  base,  and  the  shadows  of  the  horizontal 
bars  cast  upon  these  planes  are  directed  towards  these 
points ;  7,  8,  and  9  are  the  points  at  which  the  inclined 
ray  pierces  these  three  vertical  planes,  and  10  and  11 
those  at  which  it  pierces  the  two  inclined  planes,  and 
the  shadows  of  the  inclined  bars  falling  upon  these  five 
surfaces  are  directed  necessarily  towards  these  five  points 
or  apexes. 


208  MODERN   PERSPECTIVE. 

The  shadow  upon  the  vertical  wall  of  the  irregular 
line  of  the  arch  and  the  moulding  that  supports  it,  is 
found  by  the  method  shown  in  Fig.  81,^.  (311.)  Hor- 
izontal lines  are  drawn  from  successive  points  per- 
pendicular to  the  surface  of  the  wall,  and  a  line  of 
shadow  is  drawn  through  the  shadow  of  the  successive 
points. 

317.  It  is  to  be  noticed  that  if  one  were  to  stand  under 
the  arch,  with  his  eye  at  the  point  occupied  by  tlie  lamp, 
he  would  see  just  so  much  of  the  wall  as  is  now  in  light. 
These  processes  enable  us  then  to  determine  how  much 
of  wdiat  is  shown  in  a  picture  is  visible  to  the  per- 
sonages represented  in  it.  In  Fig.  83,  for  example,  if 
we  suppose  the  eyes  of  the  young  man  standing  outside 
the  arch  to  be  an  apex  of  visual  rays,  and  then  find  the 
shadow,  so  to  speak,  cast  by  them  upon  the  wall,  the  point 
12  being  the  projection  of  this  apex  upon  the  plane  of 
the  wall,  we  find  that  the  light  of  his  eyes  just  reaches 
the  young  woman  sitting  at  the  window  within. 

The  most  frequent  examples  of  shadows  cast  by  arti- 
ficial light  are  presented  in  pictures  of  interiors  lighted 
by  candles  or  gaslight.  It  is  customary  to  restrict  the 
discussion  of  these  phenomena  to  cases  in  which  the 
source  of  light  is,  as  in  the  figures  already  given,  in 
front  of  the  spectator.  But  the  theory  of  false  apexes 
explained  in  this  chapter  (301)  enables  us  to  illustrate 
also  the  case,  equally  common  in  actual  experience,  in 
which  the  source  of  light  is  behind  the  spectator.     Figs. 


SHADOWS    BY   ARTIFICIAL    LIGHT.  209 

85  and  86  illustrate  these  two  cases.     The  shadows  are 
cast  by  the  method  of  Fig.  81,/ and  g. 

318.  Figs.  84  and  85,  Plate  XIX.,  represent  the  plan 
of  a  long  room,  and  the  perspective  of  one  end  piatexix. 
of  it.  The  line  ij}),  in  Fig.  84,  shows  the  posi-  and  85. 
tion  of  the  plane  of  the  picture,  S  is  the  horizontal 
prospective  of  the  station-point,  and  A  and  B  the  pro- 
jection of  two  gas-burners  suspended  from  the  ceiling. 
The  burner,  A,  appears  in  its  proper  position  in  Fig. 
85 ;  its  projection  upon  the  floor  is  shown  at  A^,  upon 
the  ceiling  at  A^,  on  the  right  and  left  hand  walls, 
at  A^  and  A^,  and  upon  the  rear  wall,  which  is  parallel 
with  the  plane  of  the  picture,  at  A^.  These  points  are 
the  points  in  each  surface  towards,  or  away  from,  which 
the  shadows  of  lines  normal  to  these  surfaces  are  di- 
rected when  this  burner  is  lighted  (314).  Each  is  an 
apex  of  the  diverging  visible  shadows,  while  A  is  the 
apex  of  the  diverging  lines  of  invisible  shadow.  The 
extreme  point  of  tlie  visible  shadow  of  the  normal  line 
is  in  each  case  determined  by  the  line  of  the  invisible 
shadow  of  the  extreme  point  of  that  line. 

In  Fig.  86    the  other  burner,  B,  is  supposed  to  be 
lighted ;  and  we  have  the  case  already  illus- 

^  '  ^  Fig.  86. 

trated  in  Figs.  77  and  80,  in  which  the  apex 
of  diverging  lines  is  behind  the  spectator.  Following 
the  method  of  Fig.  80,  we  find  that  lines  drawn  normal 
to  the  plane  of  the  picture  from  these  several  apexes 
pierce  the  picture  at  the  points  B^j,  B§,  B|,  B§,  and  B^, 
and  the  false  apexes  B',  B^',  W,  B^,  and  B^'  are  then 
easily  found. 

14 


210  MODERN   PERSPECTIVE. 

B'  is  determined  by  the  aid  of  the  plan,  Fig.  84,  the 
The  false  ^^^^^  ^^^  being  the  perspective  of  the  ray 
apexes.  normal  to  the  picture,  and  the  vertical  line 
through  b  the  perspective  of  the  horizontal  ray  passing 
above  the  spectator.  Their  point  of  intersection,  B',  is 
the  false  apex  (303).  It  lies  in  the  shadow  of  the  spec- 
tator's head,  just  at  the  point  occupied  by  his  right  eye, 
the  eye  with  which  he  is  supposed  to  be  looking.  This 
point  is,  then,  in  artificial  light  as  well  as  in  sunlight, 
the  focus  of  rays  of  light  proceeding  from  behind 
the  spectator.  Lines  drawn  in  like  manner  from  the 
points  B§,  B|,  etc.,  through  the  centre,  C,  will  con- 
tain the  other  false  apexes,  and  as  the  true  apexes  are 
either  on  a  level  with  B  or  are  directly  above  or  below 
it,  so  the  false  apexes  will  be  either  on  a  level  with  B' 
or  directly  below  or  above  it ;  that  is  to  say,  they  will 
lie  in  the  intersection  of  the  lines  drawn  through  C, 
with  horizontal  or  vertical  lines  drawn  throuoh  B'. 

319.  It  will  be  noticed  that  the  apex,  B^,  towards 
which  the  shadows  of  the  lines  normal  to  the  rear  wall 
are  directed,  occupies  the  same  position  as  the  point  A^, 
towards  which  the  shadows  of  the  same  lines  were  di- 
rected in  Fig.  85.  The  only  difference  in  the  two  cases 
is  that  the  shadows  in  Fig.  86  are  shorter  than  those  in 
Fig.  85,  being  cut  off  by  lines  converging  to  B'  instead 
of  by  lines  converging  to  A;  but  they  coincide  with 
them  as  far  as  they  go.  A^  and  A  being  near  together, 
the  lines  directed  towards  them  intersect  a  good  way 
off,  giving  longer  shadows  tlian  do  the  lines  directed 
towards  B^  and  B^ 


THE   SHADOWS    OF   COJsWERGENT   LINES.  211 

Indeed,  there  is  mucli  less  of  shadow  altogether  in 
Fig.  86  than  in  Fig.  85,  as  was  to  be  expected,  the  spec- 
tator's eye  being  so  nearly  in  a  line  with  the  source  of 
lio-ht.  If  it  could  coincide  with  it,  as  was  supposed  to 
happen  in  the  case  of  the  serenader  in  Fig.  83,  he  would 
see  no  shadows  at  all. 

The  shadow  of  the  lantern,  A,  thrown  upon  the  rear 
wall  at  A^,  or  B^,  is  really,  of  course,  a  little  larger  than 
the  lantern,  being  at  a  greater  distance  from  the  apex,  B. 
But  its  perspective  is  smaller  than  the  perspective  of 
the  lantern,  its  distance  from  the  station-point  being 
relatively  greater  still.  If  the  station-point  were  ex- 
actly on  a  line  between  the  point  B  and  the  lantern,  the 
outline  of  the  shadow  would  obviously  fall  within  that 
of  the  lantern  itself. 

320.  The  shadows  of  convergent  lines,  whether  cast  by 
natural  or  by  artificial  light,  converge,  of  course,  shadows  of 
to  the  shadow  of  their  apex.  But  if  the  apex  miTbfrrti- 
is  just  as  far  from  the  plane  on  which  the 
shadows  fall  as  is  the  source  of  light,  its  shadows  will 
be  at  an  infinite  distance,  and  the  shadows  of  the  con- 
verging lines  will  be  parallel  to  one  another  aiid  to  the 
shadow  of  that  element  of  the  convergent  system  of  lines 
which  is  normal  to  the  plane,  and  will  have  the  same 
vanishing-point  in  the  trace  of  that  plane ;  if  it  is  far- 
ther off,  they  will  diverge,  being  directed  towards  a  point 
in  the  plane  beyond  the  source  of  light.  This  point  is, 
so  to  speak,  the  negative  shadow  of  the  apex,  being  still 
the  point  where  the  line  through  the  apex  of  the  lines 
and  the  apex  of  the  rays  pierces  the  plane.     It  is  the 


212  MODERN   PERSPECTIVE. 

same  point  that  would  be  obtained  if  the  two  apexes 
should  change  places. 

321.  Fig.  87  illustrates  these  points,  showing  the 
shadows  of  several  geometrical  figures  cast 
upon  the  floor.  The  point  A°  is  the  horizon- 
tal projection  of  the  gas-light,  A.  The  shadows  of  the 
lines  which  meet  at  the  apex  B,  meet  at  its  shadow,  B^ 
The  shadows  of  the  vertical  axes  are  directed  upon  A^. 
The  apexes  C  and  C,  are  just  on  a  level  with  A ;  their 
shadows  upon  the  table  are  accordingly  at  an  infinite 
distance,  and  tlie  shadows  of  the  lines  that  meet  at  these 
apexes  are  accordingly  parallel  in  space,  and  have  their 
vanishing-points  upon  the  Horizon.  Eacli  point  D  is 
further  from  the  table  than  A,  and  the  shadows  of  the 
edges  of  the  pyramid  of  which  it  is  the  apex  converge 
towards  the  corresponding  point  D',  the  "  negative 
shadow"  of  D,  being  the  point  upon  the  horizontal 
plane  where  the  shadow  of  A  would  fall  if  D  were  a 
source  of  light. 

In  sunlight  also  the  shadows  of  converging  lines  are 
parallel  when  the  source  of  lioht  is  iust  as  far 

Fig.  88.       ^  "^  -^ 

as  the  apex  from  the  plane  on  which  the 
shadow  is  cast.  Fig.  88  shows  how  the  Pyramids  at 
sunrise  throw  parallel  lines  of  shadow  towards  the 
western  horizon. 

The  analogy  of  these  phenomena  of  shadow  to  those 
of  the  perspective  representation  of  converging  lines 
(301,  Fig.  77)  recalls  the  analogy,  already  pointed  out,  be- 
tween the  shadows  of  all  objects  when  cast  upon  a  plane 
by  artificial  light,  and  their  perspective  representation 
(265,  Fig.  60). 


CHAPTEE  XV. 

OTHER   SYSTEMS    AND   METHODS. 

IN  the  processes  hitherto  described  every  line  has 
been  regarded  as  a  portion  of  an  infinitely  long 
line  tending  towards  its  vanishing-point,  and  eveiy 
surface  as  a  portion  of  an  infinite  plane  extending  to  its 
trace,  or  horizon ;  and  it  is  by  determining  the  position 
of  these  vanishing-points  and  horizon  that  the  position 
of  the  perspective  representations  of  these  lines  and  sur- 
faces has  been  fixed.  This  way  of  looking  at  the  sub- 
ject involves  a  comprehensive  survey  of  the  phenomena 
in  question,  and  leads  to  a  proper  understanding  of 
their  relations.  The  processes  deduced  from  this  study 
are  also  generally  convenient  in  practice ;  for,  tliough 
some  of  the  vanishing-points  are  generally  somewdiat 
remote,  still  the  space  required  for  drawings  executed 
upon  the  small  scale  commonly  employed  is  not  greater 
than  can  usually  be  afforded. 

Before  dismissing  the  subject,  however,  it  is  proper  to 
consider  some  other  methods  of  obtaining  the  same  results, 
based  upon  the  consideration  of  these  same  phenomena, 
and  involving  a  more  extended  application  of  some  of  the 
principles  already  considered,  —  methods  which  under 
certain  conditions  offer  considerable  advantages. 


214  MODERN   PEllSrECTlVE. 

322.  Several  of  these  special  methods  are  illustrated 
Plats  XX.  in  Plate  XX.  In  all  of  them  tlie  considera- 
point's  dis-      tion  of  vanishino'-points  and  horizons  is  more  or 

used ;  space 

economized.  Jess  dispensed  with,  the  lines  to  be  represented 
being  considered  merely  as  finite  lines  lying  between 
two  points,  the  immediate  object  of  the  processes  em- 
ployed being  to  fix  the  perspective  of  these  points.  In 
some  of  these  methods  the  abandonment  of  the  outlying- 
vanishing-points  leads  to  so  great  a  reduction  of  the 
space  required  for  making  the  drawing  that  the  work  is 
performed  almost  entirely  within  the  limits  of  the  pic- 
ture itself.  In  executing  large  works,  such  as  scene 
paintings  or  mural  decorations,  this  is,  obviously,  of 
great  convenience. 

The  Method  of  Direct  Projection. 

323.  In  this  method  no  use  at  all  is  made  of  vanish- 
The  method  iiig-points,  and  no  reference  is  had  to  any  of 
ka'fpJoje^c^'^'  the  phenomena  of  parallel  lines  that,  are  •  rep- 
resented by  means  of  them. 

The  object  to  be  represented  is  carefully  drawn,  both  in 
plan  and  in  side  elevation,  and  the  plane  of  tlie  picture, 
seen  edgewise  or  in  section,  and  the  station-point  are 
shown.  By  drawing  lines,  representing  the  visual  rays, 
from  every  point  in  the  object  to  the  station-point,  first 
on  the  plan  and  then  in  the  elevation,  and  noting  tlieir 
intersection  with  the  plane  of  the  picture,  the  horizontal 
and  vertical  position  of  the  perspective  of  every  point 
may  be  ascertained,  and  a  representation  of  the  object 
obtained  by  drawing  lines  connecting  tlie  points. 


THE   COMMON   METHOD.  215 

Fig.  89  illustrates  this  method,  giving  at  a,  h,  and  c, 
the  plan  of  a  cross,  set  obliquely,  and  two  ele- 

.  .  Fig.  89. 

vations,  both  of  which  are  necessary,  as  neither 
one  of  them  exhibits  all  the  points  visible  from  the 
station-point,  S,  in  front.  Lines  representing  the  visual 
rays  are  drawn,  both  in  plan  and  in  both  elevations, 
from  all  the  visible  points  to  the  point  S,  and  the  points 
where  they  pierce  the  plane  of  the  picture,  |?^,  indi- 
cated. These  points,  being  transferred  to  the  side  and 
bottom  lines  of  tlie  figure,  89,  d,  suffice  to  determine  the 
position  of  each  point  in  perspective. 

This  kind  of  projection,  in  which  the  lines  of  projec- 
tion converge  to  a  point,  instead  of  being  parallel  as  in 
plans  and  elevations,  is  called  Conical  Projection,  as 
distinguished  from  Orthographic. 

The  Mixed,  or  Common  Method. 

324  The  method  of  Direct  Projection  is  seldom  used 
to  determine  vertical  dimensions,  —  that  is  to  ^j^^  common 
say,  to  fix  the  position  of  horizontal  lines,  —  m^^ioa. 
the  labor  of  constructing  two  oblique  elevations  being 
intolerable ;  but  it  is  very  generally  employed  for  the 
determination  of  horizontal  dimensions,  —  that  is,  to 
fix  the  position  of  vertical  lines,  the  length  of  vertical 
lines  being  determined  by  means  of  lines  of  vertical 
measures  and  vanishing-points  on  tlie  Horizon. 

Fig.  90  illustrates  the  application  of  this  mixed 
method  to  the  subject  of  the  previous  fio'ure. 

.  Fig.  90. 

The  vertical  lines  are  drawn  as  in  Fig.  89,  d, 

their  position  being  taken  from  the  geometrical  plan 


216  MODERN   PERSPECTIVE. 

at  a,  by  direct  projection.  Their  length  is  determined 
by  setting  off  the  real  heiglits,  as  given  by  the  eleva- 
tion alono-side,  on  a  line  of  vertical  measures,  v  v,  taken 
where  the  plane  of  the  front  of  the  cross  intersects  the 
plane  of  the  picture.  This  is  fixed  by  the  point  m,  in 
rig.  89,  a.  Fig.  89  also  serves  to  determine  the  vanisli- 
ing-points  Y^  and  V^,  and  the  corresponding  points  of 
distance,  D^  and  D^ 

325.  Though  this  method  is  deficient  in  scientific 
The  advan-  ^^^^^7'  ^^  entirely  different  principle  being 
orthographic  cmploycd  for  horizontal  dimensions  from  that 
^^^'  used  to  determine  vertical  dimensions,  it  is 
often  very  convenient  in  practice,  especially  when,  as  is 
frequently  tlie  case  with  buildings,  a  carefully  drawn 
ground-plan,  prepared  for  other  purposes,  can  be  taken 
advantage  of  This  is  still  the  process  most  commonly 
employed  by  architectural  draughtsmen  for  the  determi- 
nation at  least  of  the  main  lines  of  their  work.  Points 
of  distance,  points  of  measures,  and  the  vanishing-points 
of  inclined  lines,  are  employed,  if  employed  at  all,  only 
as  auxiliaries  and  alternative  devices. 

326.  But  the  employment  of  the  Perspective  Plan  to 
^,     ^         determine  horizontal   distances,  and  thus  fix 

The  advan- 

ptSpe^^c^ve  the  position  of  the  vertical  lines  of  a  per- 
^'*°'  spective   drawing,   as  has  been  done  in  the 

previous  chapters,  is  altogether  preferred  by  the  best 
and  most  recent  writers,  and  by  the  best  informed 
draughtsmen.  It  has  the  signal  advantage  of  avoiding 
the   confusion   and   error   that   necessarily  attend   the 


THE    COMxMON    METHOD,  217 

multiplication  of  points  of  intersection  distributed  along 
a  single  line.  Even  in  the  figure  just  given,  simple  as 
it  is,  we  find  in  89,  «,  a  dozen  points  crowded  together 
upon  the  line  2^P-  It  is  not  easy,  in  transferring  them 
to  Fig.  90,  to  keep  clearly  in  mind  which  is  which, — 
which  indicates  a  point  at  the  bottom,  which  a  point 
on  the  top,  which  belong  to  the  front  plane  and  which 
to  the  back.  In  the  perspective  plan,  on  the  contrary, 
Fig.  91,  every  point  is  significant;  there  is  no 

Fig.  91. 

confusion,  and,  the  relations  of  all  the  parts 
being  clearly  exhibited,  there  is  much  less  danger  of 
trivial  inaccuracies  than  in  a  blind  and  merely  mechani- 
cal procedure.  Moreover,  if  the  perspective  plan  itself 
becomes  too  crowded  with  details,  it  is  practicable  to 
make  a  second  or  a  tliird,  as  has  already  been  done  in 
Plate  III.  In  the  case  of  high  buildings  it  is  usual 
to  make  a  separate  perspective  plan  for  each  story,  those 
of  the  upper  stories  being  drawn  above  them,  as  those 
of  the  lower  stories  are  drawn  below.  These  plans  are 
always  perfectly  intelligible  and  serviceable  after  any 
lapse  of  time,  and,  as  has  been  said,  may  conveniently 
be  made  on  separate  strips  of  paper,  tlms  saving  the 
drawing  itself  from  disfigurement,  and,  indeed,  protect- 
ing it  from  injury.  These  strips  of  paper  with  the  plans 
upon  them  can  then  be  preserved,  and  in  case  a  second 
drawing  for  any  reason  has  to  be  made,  half  the  labor  of 
making  it  will  have  been  saved. 

327.  Other  and  incidental  advantages  of  this  method 
are  the  great  facilities  it  offers  for  designing  in  perspec- 
tive, for  working  up  a  perspective  drawing  from  rough 


218  MODERN   PERSPECTIVE. 

sketches,  and  altering  and  adding  to  it  at  will,  studying 
the  effect  of  such  changes  as  may  be  suggested  by  taste 
or  convenience.  It  is  also  to  be  observed  that  the 
perspective  plan  takes  up  less  room  than  the  ortho- 
graphic plan,  with  its  system  of  visual  rays  directed 
towards  the  station-point,  and  this  is  sometimes  a 
consideration  of  some  importance. 

328.  The  reason  why  the  perspective  plan  is  so  little 
Sinking  the     uscd,  although  the  theory  of  points  of  distance 

perspective  .    ,       .       .  ,     . 

plan.  on  which  it  is  based  is  perfectly  familiar,  is 

that  unless  this  plan  is  sunk  considerably  below  the 
picture  the  desired  points  are  not  very  accurately  ascer- 
tained, the  lines  whose  intersections  determine  them 
cutting  each  other  at  an  acute  angle.  Sinking  the 
plan,  however,  as  is  done  in  Fig.  90  and  elsewhere,  en- 
tirely obviates  this,  and  has  the  advantage  not  only 
of  enabling  one  to  draw  it  on  a  separate  paper  and 
preserve  it  for  future  use,  as  has  just  been  suggested, 
but  of  keeping  the  picture  itself  free  from  construction 
lines. 

The  Method  of  Co-ordinates. 

329.  The  method  of  Co-ordinates  applies  the  prin- 

ciples of  Parallel  Perspective,  as  set  forth  in 

The  method  ^  ^ 

JlnSr'co-  Ciiapter  A' II,  to  tlie  solution  of  every  class  of 
ordinates.  problems.  Liiics  parallel  and  perpendicular  to 
the  picture  are  treated  as  is  usual  in  that  system.  Lines 
inclined  to  the  picture  are  determined,  as  in  the  method 
of  Direct  Projection,  by  ascertaining  the  perspective  of 
the  points  between  which  they  lie,  their  vanishing-points 


THE    METHOD    OF    CO-ORDINATES.  219 

being  neglected.  Tlie  position  of  a  point  in  space  being 
known,  the  three  dimensions  that  determine  its  position 
can  easily  be  put  into  perspective,  two  of  them  being 
taken  parallel  to  the  picture,  and  the  third  perpendicular 
to  it ;  and.  tlie  perspective  of  every  point  being  thus 
ascertained,  the  lines  lying  between  them  are  easily 
drawn. 

In  speaking  of  these  three  directions,  at  right  angles 
to  each  other,  it  is  convenient,  just  as  w^e  call  Height, 

.-IT  •  Tx    •    1  width,  and 

the  vertical  dimension  Height,  to  speak  of  the  depth, 
horizontal  dimension  parallel  to  the  picture  as  Width, 
or  Breadth,  and  of  the  otlier  horizontal  dimension,  per- 
pendicular to  the  picture  and  parallel  to  the  Axis,  as 
Depth. 

330.  Fig.  92  exhibits  the  application  of  this  method 
to  the  same  subject  as  that  by  wdiicli  the  other  ^.  ^^ 
methods  just  mentioned  were  illustrated.  The  " '""^  *' 
eye  being  supposed  to  be  about  two  inches  from  the 
paper,  the  point  of  distance  would  be  two  inches  from 
V*^,  the  centre  of  the  picture.  The  point  of  half-distance 
is  accordingly  set  one  inch  off,  at  DJ,  and  the  perpen- 
dicular dimensions  are  laid  off  upon  the  ground-line  of 
the  perspective  plan  in  Fig.  92,  h,  at  half  the  scale  of 
the  orthographic  plan  above  (Fig.  92,  a),  from  which 
they  are  taken.     (142.) 

In   Fig.  92,  c,  tlie  vertical  dimensions,  as  given  by 
tlie  elevation   in    Fig.  90,  are  laid  off  upon  vertical 

dimensions, 

the  scale  of  heights  erected  at  g.     Horizontal  Fig.  92,  c. 
lines  drawn  from  the  points  thus  ascertained  to  the 
centre,  V^,  and  vertical  lines  drawn  from  the  points  pre- 


220  MODERN   PERSPECTIVE. 

viously  ascertained  upon  the  scale  of  depths,  drawn  from 
g  to  V^,  in  the  plan  below,  determine  by  their  intersection 
the  height  above  the  ground-plane  and  the  distance 
The  side        behind  the  plane  of  the  picture  of  every  point 

elevation  in  . 

perspective,  in  the  objcct  to  bc  represented,  ihis  enables 
one,  if  he  pleases  to  do  so,  to  construct  a  perspective  of 
the  side  elevation,  as  is  done  in  the  figure,  just  as  the 
perspective  of  the  plan  is  constructed.  In  fact,  Fig.  92,  c, 
is  the  perspective  of  Fig.  89,  h,  just  as  Fig.  92,  a,  is  the 
perspective  of  Fig.  89,  a.  The  perspective  plan  and  ele- 
vation being  both  given,  the  perspective  of  the  object  is 
easily  constructed. 

It  is  sometimes  convenient  to  construct  this  perspec- 
tive elevation  in  a  vertical  plane  not  perpen- 

Fig.  92,  (^.         ,  .  . 

dicular  to  the  picture ;  a  plane,  that  is,  whose 
horizontal  elements  are  directed  to  some  other  point  of 
the  horizon  than  the  centre,  Y^.  This  is  shown  in  Fig. 
92,  d.  In  this  case  points  upon  a  new  line  of  depths 
are  taken  across  from  the  line  g  Y^. 

331.  All  this,  though  simple  in  theory,  is  laborious 
in  practice,  as  the  application  of  general  methods  to 
special  problems  is  apt  to  be.  In  most  cases  it  is  not 
worth  while  to  give  up  the  facility  and  accuracy  af- 
forded by  the  use  of  vanishing-points  for  this  tiresome 
and  roundabout  process ;  but  when  the  object  to  be 
drawn  is  irregular  in  shape,  or  bounded  by  curved  lines, 
so  that  it  has  to  be  put  in  by  points  at  any  rate,  the 
method  of  rectangular  co-ordinates,  according  to  par- 
allel perspective,  best  meets  the  case.  Even  when  such 
objects  occur  in  a  drawing  made  in  angular  perspective 


THE    METHOD    OF   CO-ORDINATES.  221 

it  is  often  convenient  to  employ  it.  When,  finally,  the 
scale  of  the  drawing  is  so  large,  or,  what  conies  to  the 
same  thing,  the  space  to  work  in  is  so  small,  that  the 
vanishing-points  are  inaccessible,  this  method  is  of 
oreat  service.  By  employing  points  of  half  distance, 
or  quarter  distance,  etc.,  the  necessary  constructions  can 
generally  be  confined  within  the  limits  of  the  picture 
itself 

332.    The  most  common  application  of  the  principle 
of  co-ordinates  is  to  the  determination  of  the  The  scales  of 

height,  width, 

size  of  miscellaneous  objects,  such  as  trees,  and  depth, 
animals,  or  human  figures  in  landscapes.  A  vertical 
scale  being  established  in  the  plane  of  the  picture,  rest- 
ing upon  the  ground-line,  lines  converging  to  any  con- 
venient point  on  the  Horizon  suffice  to  show  how  large 
any  object,  a  human  figure,  for  instance,  should  be  drawn 
in  any  part  of  the  picture. 

This  use  of  a  scale  of  heights  is  illustrated  in  Fig.  93. 
The  figures  are  supposed  to  be  all  of  the  same     ^.„  93 
height  as  the  one  in  the  immediate  foreground.  Figures  etc., 

o  c"  m  a  land- 

The  scale  of  lieights,  on  the  left,  shows  how  ^''^'^■ 
tall  such  a  figure  will  appear  at  every  point  of  the 
horizontal  plane  between  the  ground-line  and  the  Hori- 
zon. The  position  of  such  a  figure  above  or  below  that 
plane  will  not  of  course  affect  its  apparent  size.  The 
man  in  the  balcony,  on  the  right,  for  instance,  is  drawn 
just  as  tall  as  the  man  on  the  platform  beneatli,  and  the 
persons  upon  the  inclined  plane  descending  to  the  water 
are  of  the  same  height  as  those  upon  the  pavement 
alongside. 


222  MODERN    PERSPECTIVE. 

The  size  of  the  different  vessels  is  determined  in  a 
similar  way. 

333.  It  is  worth  while  here  to  point  out  that  though 
To  draw  a  poiuts  of  half-distauce,  quarter-distance,  etc., 
liSufpttnts  ii^  Parallel  Perspective,  do  not  serve,  as  do 
tance,quar-    poiuts  of  dlstaucc,  as  vauishiug-poiuts  of  lines 

ter-distance,  c     i  ~o  it  ii  'ii 

etc.  or  4t>  ,  such  lines  can  nevertheless  easily  be 

Fig.  94.         drawn  through  any  point  by  their  aid. 

Let  a  and  h  in  Fig.  94  be  two  points  through  which 
it  is  desired  to  draw  lines  making  45°  with  the  axis 
and  with  the  ground4ine,  the  centre,  V^,  and  the  point 
of  half -distance,  D^,  being  given.  By  drawing  through 
these  points  lines  directed  towards  V^'  and  D|,  crossing 
them  with  a  line  parallel  to  the  horizon,  and  then 
doubling  upon  this  line  the  distance  intercepted,  lines 
may  be  drawn  which  are  obviously  directed  towards 
D=V\ 

If  the  point  of  one  third-distance  is  given,  the  inter- 
cepted portion  must  be  trebled,  as  at  c,  or  quadrupled, 
as  at  d,  if  the  point  of  quarter-distance  is  used. 

It  is  hardly  necessary  to  explain  how  a  square  can  be 
erected  on  a  given  line  parallel  to  the  ground- 
square,         line,  as  is  shown  in  Fig.  95,  using  points  of 
Fig.  95.         ^^if^  third,  and  quarter  distance. 

The  Method  of  Squares. 
334.    The  processes  of  the  method  of  co-ordinates  may 
Squaring.       ^^  much  simplified,  especially  in  the  case  of 
objects  irregular  in  plan,  by  adopting  the  de- 


THE  METHOD  OF  SQUAKES,  223 

vice  of  squaring,  commonly  used  by  draiiglitsinen  to 
assist  them  in  copying  the  outlines  of  drawings,  espe- 
cially such  as  are  to  be  copied  on  an  enlarged  or  reduced 
scale.  It  consists  in  first  covering  the  drawing  to  be 
copied  with  a  network  of  lines,  then  reproducing  tliis 
network  at  the  scale  required,  and  finally  in  filling  in, 
by  the  eye,  the  portion  of  the  drawing  included  in  each 
of  the  reticulations. 

335.  The  Method  of  Squares  applies  a  similar  pro- 
cedure to  the  putting  into  perspective  of  a  complicated 
perspective  plan.  A  network  of  lines  being  first  drawn 
over  the  plan  in  question,  its  perspective  representa- 
tion is  easily  drawn  in  parallel  perspective.  The  de- 
tails of  the  plan  can  then  be  filled  in  with  sufficient 
accuracy,  and  the  vertical  dimensions  obtained  from  a 
scale  of  heights. 

Fig.  96  illustrates  this  procedure,  a  being  the  ortho- 
graphic plan,  squared,  h  the  perspective  plan, 
and  c  the  drawing. 

The  figure  does  not  show  liow  the  heights  are  ob- 
tained. They  may  be  obtained  either  by  sciuarmg  a 
side  elevation  and  putting  it  in  perspective,  after  the 
manner  of  Fig.  92,  c,  or  by  erecting  lines  of  vertical 
measures  at  convenient  points  in  the  plane  of  the 
picture,  as  in  Fig.  90. 

336.  If  a  sunk  perspective  plan  is  used,  as  in  the 
drawing,  the  outlines  of  the  plan  in  the  picture  can 
most  easily  be  found  by  the  use  of  proportional  divi- 
ders, the  distances  of  the  corresponding  points  from  the 
horizon  being  proportional. 


224  MODERN   PERSPECTIVE. 

Mr.  Adhemars  Mctliod. 

337.  The  system  of  Mr.  Adhemar  is  an  ingenious 
variation  of  the  method  of  Co-ordinates.  Like  that 
method,  it  does  not  rely  upon  the  use  of  any  vanishing- 
Mr  Adh.5-  poiiits  except  the  one  at  the  centre  of  the  pic- 
mar's  system.  ^^^^^^  ^^^^  -^  euablcs  the  work,  if  necessary,  to 
be  entirely  confined  within  the  limits  of  the  picture 
itself.  Such  vanishing-points  as  lie  within  these  limits, 
however,  whether  they  belong  to  any  of  the  lines  that 
occur  in  the  objects  represented,  or  are  merely  aux- 
iliary, like  points  of  proportional  measures  or  points  of 
half-distance  or  quarter-distance,  are  made  the  most  of 

Tliis  system,  like  that  described  in  the  previous  para- 
graphs, is  especially  adapted  to  cases  in  winch,  from  the 
scale  of  the  drawings,  from  the  limitation  of  the  space  at 
command,  or  from  the  great  distance  of  the  station-point 
from  the  picture,  the  vanishing-points  of  the  principal 
lines  are  inaccessible.  It  leaves  one  free  to  take  the 
point  of  view  most  conducive  to  the  desired  result,  with- 
out considering  whether  the  making  of  the  drawing  will 
be  more  or  less  difficult. 

338.  In  the  application  of  this  method  Mr.  Adhemar 
employs  four  special  devices.  These  are  Small  Scale 
Data,  Vertical  Margins,  Auxiliary  Directrices,  and  the 
Inclined  Perspective  Plan. 


339.  It    is   obvious    that    the    larger  a   perspective 

„    ,,     ,  drawincf  is  to  be  made,  the  more  convenient 

Small  scale  "                                           ' 

^***'  will  it  be  to  draw  out  the  data  from  which  it 


SMALL-SCALE   DATA.  225 

is   to   be   constructed  at  a  reduced  dimension,  a  half, 

a  third,  or  a  fourth,  of  that  employed  in  the  picture. 

Instead,  however,  of  magnifying  the  data  f ur- j^^^^^^^^  ^^^^^^ 

nished  by  such  orthographic  drawings  in  order  ^/ijuTS an "'^ 

to  bring  them  up  to  the  scale  of  the  plane  of  pia^'neo? meas- 
ures, 
measures,  as  is  done  with  Fig.  84,  an  auxiliary 

plane  of  measures  is  emi^loyed,  two,  three,  or  four,  times 

as  far  off,  a  plane  so  distant  that  the  small  Rfduced  scale 

,  , .  .  .  '11  1       ^^  depth  and 

scale   dimensions   o-iven   m   the  data  can   be  fractional 

^  points  of 

employed  without  change  to  establish  scales  distance. 
of  height  and  breadth.  The  measures  of  depth,  per- 
pendicular to  the  plane  of  the  picture,  are  laid  off 
upon  the  ground-line,  as  usual ;  but  here,  too,  the  same 
small  scale  is  employed,  the  dimensions  being  trans- 
ferred to  the  scale  of  depth  by  means  of  fractional 
distance-points  (142). 

340.  Fig.  97,  a,  h,  and  c,  illustrates  this  procedure. 
At  a  we  have  first  the  elevation  of  the  object  piatexxi. 
to  be  drawn,  in  this  case  a  pyramid;  the  line,  ^'s- ^• 
of  the  horizon  is  drawn  to  show  how  mucli  of  the  pyra- 
mid is  above,  how  much  below,  the  eye.  Alongside  is  an 
Orthographic  Plan,  on  the  same  scale,  sliowing  tlie  posi- 
tion of  the  pyramid  relatively  to  the  plane  of  the  picture 
and  to  a  line,  G  0,  drawn  upon  the  horizontal  plane  per- 
pendicular to  the  picture,  at  a  convenient  distance  from 
the  p^Tamid,  to  serve  as  a  scale  of  depth.  The  dotted 
lines  show  the  distance  from  tliese  two  lines  to  the  four 
angles  of  the  pyramid  which  are  visible  from  the  station- 
point.  They  are  the  horizontal  co-ordinates  of  these 
points.      The  vertical  co-ordinates  of  these  points  are 

15 


226  MODERN    PERSPECTIVE. 

shown  by  a  dotted  line  at  the  side  of  the  elevation. 
V^  is,  as  usual,  the  centre  of  the  picture  opposite  the 
eye,  at  S,  and  D^  is  the  point  of  half-distance. 

341.  At  h  we  have  this  Orthographic  Plan  put  into 

perspective,  and  the  pyramid  in  perspective 
above  it.  But  the  scale  is  doubled,  the  dis- 
tance g  c,  upon  the  ground-line  being  double  that  of 
G  C  in  the  plan.  Instead  of  using  the  ground-line, 
however,  as  a  line  of  horizontal  measures,  or  scale  of 
widths,  as  we  have  hitherto  done,  an  auxiliary  line  of 
measures,  o  m ,  is  drawn,  at  such  a  distance  behind  the 
ground-line  as  to  make  o  c  equal  to  GO.  At  o  is  erected 
a  line  of  vertical  measures,  or  scale  of  heights.  Upon 
these  lines  the  dimensions  of  width  and  heiglit  given 
in  the  plan  and  elevation  are  set  oft'  without  cliange 
of  scale.  The  dimensions  of  depth,  given  on  the  line 
G  0,  are  set  off  at  the  same  scale  upon  the  ground- 
line,  from  the  point  g,  and  are  transferred  to  the  scale 
of  depths,  g  o,  by  means  of  the  point  of  half-distance, 
D|.  If  the  scale  were  enlarged  three  times,  a  point  of 
one-third-distance,  Dl,  would  be  employed. 

342.  These  three  scales,  or  lines  of  measures,  are 
what  are  called  in  geometry  co-ordinate  axes,  and  their 
point  of  meeting,  o,  is  called  the  origin  of  co-ordinates, 
being  the  point  from  which  the  co-ordinates  of  any 
point  are  measured.  These  being  laid  off  upon  these 
axes,  as  explained  in  the  previous  paragraph,  the  posi- 
tion of  any  point  in  space  is  easily  ascertained  by 
drawing  from  the  corresponding  points  on  each  axis 
lines  parallel  to  the  other  axes,  in  each  plane  of  pro- 


SMALL-SCALE   DATA.  227 

jectioD  ;  this  gives  the  projection  of  the  point  in  each 
of  the  three  planes.  Lines  drawn  from  each  of  these 
points  of  projection,  perpendicular  to  the  plane  in  which 
it  lies,  and  parallel  to  the  other  axis,  will  meet  in  space 
in  a  point  which  is  the  perspective  required.  As  two  of 
these  axes  are  parallel  to  the  picture,  lines  parallel  to 
them  are  drawn  parallel  to  tlieir  perspectives,  while  the 
third  axis,  being  perpendicular  to  the  picture,  is,  together 
with  the  lines  parallel  to  it,  directed  to  the  centre,  Y^. 
The  perspective  of  the  point  4,  the  vertex  of  the  pyra- 
mid, is  ascertained  in  tliis  way,  its  distance  from  the 
origin,  o,  in  each  direction,  being  first  marked  upon  the 
three  axes,  or  lines  of  measures ;  its  projection  upon 
the  three  planes  is  then  found ;  and  finally  the  point 
itself  is  found  where  the  three  co-ordinates  in  space 
meet.  Tlie  point  4  is  seen  to  be  the  front  upper  right- 
hand  corner  of  a  parallelopiped,  of  which  the  origin,  o, 
is  the  lower  back  left-hand  corner. 

343.  But  it  is  obvious  that,  as  the  intersection  of  two 
of  these  lines  would  suffice  to  determine  the  point,  the 
projection  of  the  point  upon  two  planes  is  all  that  is 
required.  The  perspectives  of  the  points  1,  2,  and  3  are 
accordingly  determined  only  upon  the  horizontal  plane, 
and  upon  the  vertical  plane  on  the  left,  perpendicular  to 
the  picture. 

,  It  is  worth  wliile  to  point  out  that  in  the  perspective 
plan  the  resemblance  between  the  converging  co-ordi- 
nates, and  the  edges  of  the  pyramid  above,  is  accidental. 
These  lines  would  converge  upon  V^,  whatever  the  shape 
of  the  object  to  be  drawn. 


228  MODERN   PERSPECTIVE. 

344.  Fig.  97,  c,  shows  that  by  omitting  to  sink  the 

perspective  plan  these  operations  may  all  be 
conducted  almost  within  the  limits  of  the 
picture  itself.  It  also  shows  that  if  we  omit  the  let- 
ters and  figures,  and  draw  in  only  so  much  of  the  con- 
structive lines  as  are  actually  necessary  to  determine 
points  required,  the  work  is  simple  and  easy,  and  does 
not  fill  up  the  space  devoted  to  the  picture. 

345.  We  have  hitherto  regarded  the  plane  of  the 
picture  as  a  surface  of  indefinite  extent,  the  exact  limits 
of  the  picture  itself  being  the  last  thing  to  be  consid- 
ered. In  the  system  under  consideration,  however,  the 
Vertical  shapc  and  size  of  the  picture  are  determined 
margins.  beforehand,  its  lower  limit  being  set  at  the 
ground-line,  from  which,  on  either  side  of  the  picture, 
arise  vertical  lines,  which  are  its  maroins  on  the  ri^ht 
and  left.  The  points  where  they  stand  being  noted  on 
the  orthographic  plan,  horizontal  boundary-lines  drawn 
through  these  points  and  the  station  point  enclose  a 
trapezoidal  surface,  wliich  is  the  area  seen  in  the  picture. 
These  horizontal  boundaries,  radiating  from  the  station- 
point,  have  their  perspectives  vertical,  in  accordance 
with  the  nature  of  converging  lines  (231).  Their  per- 
spectives are  the  vertical  margins  of  the  picture,  wliich 
to  the  spectator  at  the  station  point  seem  to  cover 
and  coincide  with  the  horizontal  boundaries.  All 
points,  accordingly,  situated  on  the  horizontal  bounda- 
ries of  the  trapezoidal  area  will  appear  in  perspective 
upon  the  vertical  margins  of  the  picture.     These  ver- 


VERTICAL   MARGINS.  229 

tical  margins  are  not  necessarily  set  at  equal  distances 
from  tlie  centre,  V^. 

346.  If,  for  instance,  the  distance  of  the  centre,  Y^, 
from  one  of  the  vertical  margins  of  the  picture  being 
determined,  the  distance  of  the  station-point,  S,  in  front 
of  the  picture,  is  three  or  four  times  that  distance,  then 
the  point  of  one-third  or  one-quarter  distance,  D^J  or  D^J, 
will  fall  exactly  upon  that  vertical  margin.  If,  also, 
any  lines  of  the  plan  are  prolonged  until  they  cut  the 
boundaries  of  the  visible  horizontal  plane,  these  points 
of  intersection  will  also  appear  in  perspective  upon  the 
vertical  margins  of  the  picture.  Such  lines  may  accord- 
ingly be  put  into  perspective  without  finding  any  points 
except  just  at  the  edges  of  the  picture,  and  upon  a  line 
of  depths. 

347.  Fig.  98  illustrates  this  procedure,  showing  how 
the  same  perspective  plan  as  that  determined 

'-         ^  ^  Fig.  98. 

in  Fig.  97  can  thus  be  obtained.  At  a  we 
have,  as  before,  a  small-scale  orthographic  plan.  As- 
suming pp',  in  the  picture  plane,  as  the  right  and  left- 
hand  limits  of  the  picture,  diverging  lines  drawn  through 
these  points  from  the  station-point,  S,  establish  the  hor- 
izontal boundaries  of  the  trapezoidal  area.  If,  now,  the 
sides  and  diagonals  of  the  plan  of  the  pyramid  be  ex- 
tended till  they  cut  these  boundaries  on  either  side  and 
the  ground-line  in  front,  we  shall  have  the  points  num- 
bered 1,  2,  3,  4,  5,  6,  7,  and  8,  their  projections  upon  the 
line  of  depths,  drawn  at  right  angles  to  p,  being  figured 
1',  2',  3',  7',  8'.  If,  now,  we  double  the  scale,  as  in  the  pre- 
vious case,  the  points  1',  2',  3',  7',  and  8',  are  easily  found 


230  MODERN   PERSPECTIVE. 

upon  the  scale  of  depths,  as  before,  and  1,  2,  3,  7,  and  8,  at 
the  same  levels  upon  the  vertical  margins.  The  points 
4',  5',  and  6/  being  taken  on  the  small-scale  line  of 
widths  serve  to  determine  the  points  4,  5,  and  6  upon 
the  ground-line. 

348.  The  perspectives  of  these  eight  points  being  thus 
ascertained,  it  is  easy,  by  drawing  the  lines  1,  6 ;  2,  7  ^ 
and  4,  8,  to  draw  the  perspective  plan  of  the  visible  half 
of  the  pyramid.  The  point  9,  the  projection  of  its  vertex, 
is  got  by  drawing  its  projection  upon  the  ground-line  in 
the  orthographic  plan.  Indeed,  in  practice,  this  method 
and  that  of  co-ordinates  are  used  interchangeably,  as  the 
conditions  of  each  case  may  render  advisable. 

349.  In  Mr.  Adhemar's  treatise  the  trapezoidal  figure, 
with  its  horizontal  boundary-lines,  is  drawn  in  the  ortho- 
graphic plan,  and  the  corresponding  vertical  margins 
are  drawn  at  the  edges  of  the  picture,  even  in  cases 
where  the  special  device  described  in  the  previous  para- 
graph is  not  employed.  They  serve  to  define  the  limits 
of  the  work,  and  it  is  often  convenient,  even  when  one 
is  employing  the  ordinary  metliod  of  co-ordinates  with 
the  origin  of  co-ordinates  in  the  plane  of  the  picture,  to 
have  the  line  of  deptlis,  i.  c,  the  line  of  perpendicular  or 
normal  measures,  and  the  ground-line,  or  line  of  hori- 
zontal measures,  meet  in  the  line  of  vertical  measures, 
which  in  that  case  coincides  with  the  vertical  margin 
in  the  corner  of  the  picture.  This  point  is  then  the 
origin  of  co-ordinates,  as  happens  in  Fig.  92.  In  em- 
ploying the  method  of  small-scale  data  also,  although 


VERTICAL   MARGINS.  231 

the  origin  of  co-ordinates  is  at  a  distance,  it  is  often 
convenient  to  have  the  ground-line,  the  line  of  depths, 
and  the  vertical  margin,  meet  at  a  point. 

350.  But  that  this  is  a  mere  matter  of  convenience, 
which  involves  no  principle,  and  really  has  no 

effect   upon  the  method  of   working  out   the  margins  not 

necessary 

perspective  problem,  may  be  seen  in  Figs.  99  ^J^^  smaii- 
and  100,  which  illustrate  the  relation  between 
the  vertical  margins  of  a  picture  and  the  system  of 
small-scale  data,  according  to  w^hich  they  are  made. 

351.  Fig.  99  shows  an  orthographic  plan,  on  a  small 
scale,  with  the  station-point,  S,  the  point  <x,  ^.  99  ^ 
of  which  the  perspective  is  to  be  found,  and  ^°'^^* 
the  plane  of  the  picture,  at  GL,  at  right  angles  to  the 
Axis,  with  the  centre  of  the  picture  at  V^,  and  the  point 
of  distance  at  D,  V^  D  being  equal  to  S  V*^.  The  line  of 
depth,  G  0'  0",  touches  the  ground-line  at  G,  and  the 
left-hand  boundary  of  the  visible  area  is  drawn  from  S 
through  this  point.  But  as  this  line  plays  no  part  in 
the  solution  of  the  problem  its  position  is  immaterial, 
and  it  might  just  as  well  have  been  directed  more  to 
the  left,  as  shown  by  the  dotted  line.  The  right-hand 
boundary  is  also  drawn  at  pleasure.  Being  drawn,  they 
fix  the  width  of  the  picture,  G  L. 

Fig.  99,  B,  shows  the  same  points  in  the  side  view, 
jy  p  being  the  plane  of  the  picture,  a  the  point  to  be  put 
into  perspective,  and  a  g  the  horizontal  plane  on  which 
it  lies,  cutting  the  plane  of  the  picture  at  the  ground- 
line,  ,9'.  Y^^,  equal  to  c  a,  shows  how  far  the  point  in 
question  lies  below  the  level  of  the  eye  at  S. 


232  MODERN   PERSPECTIVE. 

352.  Fig.  99,  C,  shows  the  picture  itself  on  a  scale 

three  times  that  of  the  plan,  V^  beino-  the 

Fig.  99,  C.  i.  o 

centre,  and  DJ  the  point  of  one-third  distance. 
The  origin  of  co-ordinates,  o,  is  found  by  setting  off  V^  ^, 
equal  to  V^  g  in  Fig.  99  B,  and  i  o,  equal  to  V^  G  in 
Fig.  99  A.  The  lines  o  z,  o  e,  and  o  g,  are  then  the 
three  co-ordinate  axes.  The  ground-line,  g  I,  is  three 
times  as  far  from  the  Horizon  as  in  Fig.  99,  B. 

353.  As  the  scale  employed  in  these  operations  is 
^^    ,       .  one-third  of   the  scale  employed   in  the  pic- 

Tne  plane  of  x       %/  l 

measures.  ^^^^.^^  -^  follows  that  the  plane  of  measures 
containing  V^  i,  o  and  e  is  three  times  as  far  from  the  eye 
as  is  the  picture.  In  the  orthographic  plan  it  lies  at  0^ 
This  distance,  G  0',  of  which  o  g  is  the  perspective,  may 
be  found  directly  by  drawing  a  line  from  DJ  through  o 
until  it  intersects  the  ground-line  at  o'.  The  distance, 
g  o\  gives  the  distance  of  o  behind  the  picture,  on  the 
scale  of  the  plan.  It  is  equal  to  G  0'.  This  gives  the 
position  of  the  plane  of  measures  (93),  the  plane  in 
which  these  dimensions  are  taken. 

354.  The  horizontal  co-ordinates  of  any  point  in  the 
perspective  plan  can  now  be  ascertained.  Let  a  be 
such  a  point.  Its  horizontal  co-ordinate,  ii  a,  equal  to 
o'  u\  being  laid  off  on  the  horizontal  axis  from  o,  gives 
the  point  e  on  that  axis,  and  the  perspective  of  the 
point  a  will  lie  in  the  normal  perspective  line  drawn 
through  c  from  the  centre, Y^.  The  point  %  upon  the 
line  of  depths,  og,  may  be  ascertained  either  by  laying 
off  upon  that  line  the  perspective  of  the  distance,  0'  u, 
from  0,  or  the  distance,  G  u,  from  g.     Either  of  these 


THE   PLANE   OF   MEx\.SUEES.  233 

distances  being  measured  upon  the  ground-line,  from 
g  or  from  o',  determines  the  point  u',  which  may  then 
be  transferred  to  the  line  of  depth,  at  it,  by  means  of 
the  point  of  distance,  D|-,  A  horizontal  line  from  u  will 
cut  the  normal  line  through  c  at  the  desired  point,  a. 

If  the  desired  point  lay  above  the  ground-plane,  its 
height  could  be  set  off  at  the  same  scale  upon  a  vertical 
ordinate  erected  at  c,  and  a  normal  line  drawn  through 
Y^,  and  the  point  thus  determined  would  cut  an  ordinate 
erected  at  a  at  the  required  point. 

355.  It  will  be  noticed  that  all  these  operations  are 
quite  independent  of  the  limits  to  be  given  to  the  pict- 
ure, and  would  have  gone  on  in  exactly  the  same  way 
if  the  vertical  margin  had  been  drawn  to  the  left  of 
the  point  g,  just  as  the  lower  margin  is  independent  of 
the  ground-line.  Moreover,  most  of  these  operations 
are  quite  independent  of  the  scale  to  be  employed, — 
that  is  to  say,  of  the  number  of  times  the  picture 
is  to  be  magnified.  This  is  determined  solely  by  the 
distance  at  which  the  ground-line  is  set  below  the 
Horizon. 

356.  This  is  illustrated  in  Fig.  99,  D,  which  is  iden- 
tical with  Ficf.  99,  C,  except  that  the  line  n  I  is 

.  .  Fig.  99,  P. 

between  five    and  six  times  as  far  from  the 
Horizon  as  is  the  horizontal  axis,  o  c,  instead  of  three 
times.     Tliis  gives  upon  the  ground-line  the  distance, 
(J  o",  and  upon  the  orthogra,phic  plan  to  point  0",  for  the 
position  of  the  plane  of  measures. 


234  MODERN   PERSPECTIVE. 

The  fractional  point  of  distance  remains  unchanged, 
and  the  point  it'  is  as  far  from  g  as  before ;  u  and  a  are 
then  easily  found.  If  the  dimensions  of  the  picture  are 
enlarged  proportionally,  as  is  here  done,  a  will  be  found 
to  occupy  the  same  relative  position  in  the  larger  pict- 
ure that  it  does  in  the  smaller  picture  above. 

It  is  not  of  course  necessary  to  know  how  large  a 
magnifying  power  one  is  employing.  The  ground-line, 
or  the  vertical  margins,  may  be  set  wherever  seems 
best.     The  rest  will  take  care  of  itself. 

357.  Fig.  100  shows  that  the  results  attained  are  also 
Fi  100  A  ^l^^ite  independent  of  the  size  and  scale  of  the 
^^^  ^'  picture  shown  in  the  orthographic  plan.     We 

have  at  A  and  B  exactly  the  same  conditions  as  in  the 
previous  figure,  except  that  the  plane  of  the  picture  is 
further  from  the  spectator,  and  nearer  the  point  a.  The 
picture  there  indicated  is  accordingly  larger,  the  point 
of  distance  further  from  the  centre,  and  the  distance, 
G  n,  taken  on  the  line  of  depth,  just  so  much  smaller. 
This  line,  being  still  dravv^n  to  the  edge  of  the  picture, 
is  further  from  the  axis,  and  the  dimension,  G  V^,  is 
greater  than  before. 

In  Fig.  100,  C,  however,  v^e  obtain  exactly  the  same 
result  as  in  Ym.  99,  C,  the  scale  of  the  en- 
larged  drawing  being  the  same. 

In  this  figure  the  ground-line  meets  the  left-hand 
vertical  margin  just  in  the  corner  of  tlie  picture.  The 
original,  at  A,  being  larger  than  in  the  previous  figure, 
does  not  need  to  be  magnified  as  many  times,  hardly 


AUXILIARY    DIKECTEICES.  235 

more  than  twice,  and  the  ground-line  is  accordingly  set 
nearer  the  Horizon. 

In  Fig.  100,  D,  the  same  result  is  a  third  time  at- 
tained, the  line  of  depth  not  beini;-  brought  to 

.  ,  .  Fig.  100,  D. 

the  corner  of  the  picture,  but  set  just  as  far 
from  the  axis  as  in  Fig.  99,  the  point  of  distance,  how- 
ever, being  the  same  as  in  Fig.  100,  C. 

358.  It  appears  from  these  examples  that,  if  it  were 
for  any  reason  convenient  to  employ  different  systems 
of  ground-lines  and  points  of  distance,  or  different  lines 
of  depth,  in  the  same  picture,  one  part  of  it  being  put 
in  by  one  system  and  another  by  another,  there  would 
be  no  discrepancy  in  the  results.  We  shall  have  occa- 
sion to  avail  ourselves  of  this  before  we  get  to  the  end 
of  the  chapter.     (375.) 

359.  We  have  seen  that  the  method  of  co-ordinates 
is  specially  useful  in  putting  into  perspective  Auxiliary 

directrices. 

irregular  figures,  figures  that  have  to  be  treated  Mouldings, 
as  a  series  of  points.  In  drawing  mouldings,  for  in- 
stance, especially  when  the  scale  of  the  drawing  is  so 
large  that  they  have  to  be  determined  with  great  pre- 
cision, the  system  of  co-ordinates  enables  us  to  fix  the 
horizontal  and  vertical  position  of  as  many  points  as  we 
please,  and  thus  to  attain  any  degree  of  exactness  in  the 
profiling  that  we  may  desire.  A  single  profile  being 
ascertained,  any  length  of  moulding  may  be  drawn  by 
drawing  lines  through  the  points  thus  determined  to  the 
proper  vanishing-points. 


236  MODERN    PERSPECTIVE. 

360.  Fig.  101  shows  bow  the  horizontal  and  vertical 

dimensions  that  define  the  outline  of  a  cornice 
may  be  determined  in  either  of  three  ways. 
At  a  is  shown  the  section  of  a  cornice,  with  the  hor- 
izontal and  vertical  co-ordinates  of  the  principal  points. 
At  h  this  section  is  put  into  perspective  by  means  of  the 
perspective  plan  K  above  and  the  point  of  distance,  D^, 
and  the  parallel  lines  of  the  cornice  are  drawn  through 
the  points  thus  determined.  At  c  the  perspective  of 
the  cornice  at  the  corner,  "  on  the  mitre,"  is  ascertained 
in  like  manner,  with  the  aid  of  the  vanishing-point  of  45°, 
V^.  This  profile  serves  for  the  cornice  on  the  right  as 
well  as  for  that  on  the  left.  At  d  the  section  parallel 
to  the  picture  is  ascertained,  and  employed  for  a  cornice 
going  off  to  the  right.  Being  used  parallel  to  the  pic- 
ture, it  is  put  into  perspective  without  change. 

Mr.  Adhemar  employs  all  these  profiles  at  once,  as 
auxiliary  to  each  other,  thus  dispensing  with  both  van- 
ishing-points, as  may  be  seen  at  e.  The  relative  posi- 
tion of  these  sections  may  be  seen  in  the  perspective 
plan  at  the  top. 

361.  This  device  is  of  special  service  in  putting  circu- 
circuiar        lar  mouldinojs  into  perspective,  as  may  be  seen 

mouldings.  .7  . 

Fig.  102.  in  Fig.  102,  which  is  borrowed,  with  the  omis- 
sion of  the  construction  lines,  from  the  plates  of  his  treat- 
ise. The  construction  of  these  profiles  is  very  easy,  and 
as  many  of  them  can  be  employed  as  seems  necessary. 

362.  In  very  large  drawings  the  use  of  auxiliary 
profiles  as  directrices  is  of  special  advantage,  because, 
even  wlien  the  vanishing-points  are  within  reach,  the 


REMOTE   OBJECTS.  237 

necessary  rulers  and  straiglit  edges  are  so  long  and  so 
flexible  that  it  is  difficult  to  keep  the  lines  true  and  the 
mouldings  properly  proportioned  throughout. 

363.  It  has  already  been  said  that  the  reason  why  the 
method  of  the  Perspective  Plan  has  attained  ^     , 

^  Remote 

so  little  vogue,  and  has  generally  been  set  °^^'^^^- 
aside  for  the  method  of  Direct  Projection,  is  this :  that 
the  lines  of  the  perspective  plan  generally  meet  at 
so  acute  an  angle  as  to  make  it  difficult  to  determine 
their  exact  point  of  intersection.  This  objection  is 
overcome  by  the  device  of  sinking  the  plan,  so  that 
the  perspective  lines  may  intersect  more  nearly  at 
right  angles.  But  however  satisfiictorily  this  may  be 
arranged  for  the  foreground  of  the  plan,  lines  in  the 
more  distant  parts  must  always  grow  gradually  more 
nearly  parallel  to  the  horizon  and  to  each  other,  so  that 
in  just  that  part  of  the  drawing  where  the  scale  is 
the  smallest,  and  precision,  accordingly,  most  important, 
precision  is  tlie  most  difficult  to  obtain. 

364.  To  remedy  this  evil  Mr.  Adhemar  proposes  that 
the  more  distant  objects,  or  the  more  distant  ^j^^  jncuned 
portions  of  an  object,  shall  be  projected  not  pfan!'''''**^® 
upon  a  horizontal  plane  but  upon  an  inclined  Fig  loi. 
plane,  thus  increasing   the  vertical   distances   'P'^*^^^^^^- 
and  the  angles  at  which  the  lines  of  the  plan  intersect 
with  one  another.     Ym.  103  shows  the  device 

^  Fig.  103,  A. 

in  question.     At  A  is  shown  tlie  station-point, 

S,  the  plane  of  the  picture,  j;  ^9',  seen  in  profile,  and  the 

elevation,  or  edge,  of  a  horizontal  ring  tangent  to  the 


238  MODERN   PERSPECTIVE. 

plane  of  the  picture  at  p,  aud  the  projection  of  this  ring 
upon  the  ground-plan  at  g  g.  This  will  of  course  be 
a  circle,  and  its  perspective,  as  seen  from  S,  will  be  an 
ellipse,  the  major  axis  of  which  is  horizontal  and  rather 
less  than  the  diameter  of  the  circle,  and  the  minor  axis 
vertical,  and  equal  to  the  distance,  dg.  This  ellipse  is 
sliown  at  a,  where  is  shown  also  the  ground-line,  g  I, 
the  centre,  C,  the  major  axis  of  the  ellipse  ^^  f  f,  and 
the  horizontal  diameter  of  the  circle,  which  appears  as 
a  horizontal  chord  of  the  ellipse  at  e  c,  the  further  half 
of  the  circle  appearing,  of  course,  smaller  than  the  nearer 
half  The  centre  of  the  circle  is  seen,  in  the  plane  of 
the  picture,  at  c. 

365.    Mr.  Adhemar's  device  for  making  the  perspec- 
tive of  the  further  half  of  tlie  circle  as  wide  as  that  of 
the  nearer  half  is  shown  at  B.     The  further  half  of  the 
ring  is  projected  not  upon  a  horizontal  but  upon  an 
inclined  plane.     This  plane  is  showm  in  the 

Fig.  103,  B. 

figure,  cutting  the  plane  of  the  picture  in  a 
new  ground-line,  at  (j ,  the  horizon  of  this  plane  being 
not  at  V*^,  but  at  V^'.  Using  this  new  horizon  and  this 
new  ground-line,  the  projection  of  the  further  half  of  the 
ring  can  be  drawn,  as  shown  at  h,  with  any  desired  pre- 
cision, the  same  as  upon  any  inclined  plane. 

But  this  can  be  equally  well  and  more  simply  accom- 
plished, merely  by  sinking  to  a  still  lower  level  the  fur- 
ther half  of  the  circle  in  which  the  ring  is  projected,  that 

is  to  sav,  by  proiectin<]f  the  further  half  of  the 
Fig.l03.C.        .  ,  .      r       1      1  1     . 

ring  not  upon  an  inclined  plane  but  upon  a 

lower  horizontal  plane,  as  is  done  in  Fig.  103,  C.     This, 


THE  SUNK  PERSPECTIVE  PLAN.         239 

to  be  sure,  tlioiigli  it  widens  the  projection  of  the  farther 
parts  as  much  as  may  be  desired,  brings  their  perspec- 
tive lower  down  than  that  of  the  nearer  parts,  as  at  c, 
which  is  awkward.  But  this  can  be  got  over,  by  not 
only  sinking  that  portion  of  the  perspective  plan,  but 
also  simultaneously  raising  the  station-point.  If  this 
is  done  judiciously,  as  at  D,  the  two  portions 

•^  *^  ^  Fig.  103,  D. 

of  the  perspective  plan  will  preserve  their  rela- 
tive position,  as  at  d.     The  next  figure  shows  how  this 
may  be  effected  (370). 

366.  The  results  reached  in  B  and  D  are  substan- 
tially alike,  and  it  is  plain  that  sinking  the-  The  sunk 

plane  pref- 

remote  portions  of  the  perspective  plan,  in-  erabieto 

^  i         i.  L  ^Y\Q  inclined 

stead  of  using  an  inclined  plane  of  projection,  p^^"^- 
accomplishes  the  practical  end  in  view,  by  what  seems 
to  be  a  simpler  method.  It  is  worth  while  to  point  out, 
however,  before  leaving  the  subject,  that  the  results  in 
the  two  cases  are  not  only  similar  but  identical ;  that  is 
to  say  that  the  perspective  of  the  projection  upon  the 
sunken  horizontal  plane  coincides  with  that  of  the  pro- 
jection upon  the  inclined  plane,  not  only  at  its  extremi- 
ties but  at  every  point. 

367.  This  may  be  seen  in  Fig.  104,  where  the  line 
A  G,  arbitrarily  divided  into  two  parts  at  the  The  results 

identical. 

point  B  is  seen  projected  both  upon  a  hori-  Fig.io4. 
zontal  plane  at  a,  h,  and  g,  and  upon  an  inclined  plane 
at  a',  V,  and  g.  The  figure  shows  that  if  the  new  sta- 
tion-point S',  immediately  above  S,  is  so  taken  that  the 
perspective  of  a  as  seen  from  S  coincides  witli  that  of 
a!  as  seen  from  S,  the  point  g  common  to  both  projec- 


240  MODERN   PEKSPECTIVE. 

tions  being  its  own  perspective,  then  the  perspective  of 
h  as  seen  from  S'  will  coincide  with  that  of  h'  as  seen 
from  S.  That  is  to  say,  if  the  perspectives  of  the  two 
projections  coincide  at  their  extreme  points  they  will 
also  coincide  at  every  intermediate  point. 

368.  The  proof  of  this  follows  from  tlie  relations  of 
the  lines  of  the  figure,  in  which  perpendiculars  let  fall 
from  the  two  station-points  S  and  S'  give  the  correspond- 
ing Horizons  at  V^,  and  V^'.  For,  in  the  first  place, 
since  the  trapezoids  SS'  Q'  a!'  and  g  a  a'  a'^  are  divided 
by  the  line  a  S^  into  similar  triangles,  the  trapezoids 
themselves  must  be  similar,  and  since  their  homologous 
sides  are  parallel,  their  diagonals  S  V^'  and  g  a'  are  also 
parallel.  The  right-angled  triangles  S  S'  V^'  and  h  V  g, 
having  their  vertices  upon  the  line  C  g  are  accordingly 
similar,  and  their  heights,  S'  V^'  and  h  g,  are  proportional 
to  their  bases  S  S'  and  h  V.  But  the  scalene  triangles 
S  S'  y^  and  h  V  h^'  are  also  similar,  and  their  heights  also 
are  proportional  to  their  bases  S  S'  and  h  V,  which  are 
the  same  as  those  of  the  rioht-anoled  trianoles.  Their 
heights  must  accordingly  Ije  the  same  as  the  heights  of 
the  right-angled  triangles,  and  their  vertices  at  V  must 
also  fall  upon  the  line  V^'  g.  That  is  to  say,  the  per- 
spective of  h'  as  seen  from  S,  coincides  with  that  of  h 
as  seen  from  S',  and  this  is  true  wherever  the  transverse 
line  h  h'  may  be  drawn. 

369.  Since,  as  has  been  shown,  S  V^'  and  g  a^  are 
parallel  whenever  S  a'  and  S'  a  intersect  in  the  plane 
of  the  picture,  it  follows  that  the  point  V^' which  marks 
the  position  of  the  Horizon  of  the  horizontal  plane  of 


SUCCESSIVE   HORIZONTAL   PLANES.  241 

projection  ah  g,  when  the  eye  is  at  S',  marks  also  the 
horizon  of  the  inclined  plane  of  projection  a'  h'  g  when 
the  eye  is  at  S,  and  that  g  marks  the  ground-line  of 
both  the  sunk  plane  and  tlie  inclined  plane. 

370.  This  enables  us  to  answer  a  question  raised  in 
resi^ard  to  Fis:.  103,  D  and  d  (367),  and  to  say 

'^  o  '  \  J^  J     To  find  C 

just  how  far  the  Horizon  must  be  raised  and  ^"^s^'- 

,  Fig.  103,  D. 

the  ground-line  lowered  in  order  to  make  the 
perspectives  of  successive  sunk  planes  continuous,  one 
with  another,  like  the  perspectives  of  successive  in- 
clined planes.  The  successive  horizons  and  ground- 
lines  of  the  horizontal  planes  are  to  be  drawn  just 
where  the  successive  horizons  and  ground-lines  of  the 
inclined  planes  would  be,  the  distance  that  tlie  station- 
point  is  raised  being  proportional  to  its  distance  in  front 
of  the  yjicture,  and  the  distance  that  the  horizontal 
plane  is  lowered  being  proportional  to  its  distance  be- 
hind the  picture. 

371.  The  identity  of  tlie  results  obtained  by  these 
two    processes    is   again    illustrated   in   Fig.  The  results 

^  7  1      7  A  -1  ^^^  "anie 

lOo,  a,  6,  c,  and  a.     At  a  is  the  orthocrraphic  ty  either 

^      ^  method. 

projection  of  a  circle,  tangent  to  the  plane  of  Fig.  los.  a. 
tlie  picture,  G  L,      The  area  of  this  circle  is  divided  into 
four  segments  by  planes  which  cut  the  line  of  depth  at 
the  points   A,  B,  D,  E,  and  G.     In  the  Fig.  rig.  los,  c 
105,  c,  above,  these  seo-ments  are  shown  pro-  f»^5'<'^«i/e 
jected  upon  four  horizontal  planes  having  their  ^^''°^^' 
ground-lines  at  g,  g"  g'",  and  ^^^,  the  corresponding  sta- 
tion-points,   S,  S'',  S'",  and  S^^  being  so  adjusted,  in 

16 


242  MODERN    PERSPECTIVE. 

accordance  witli  the  relations  shown  in  Fig.  104,  that 
the  perspectives  of  the  plans  of  the  different  segments 
form  a  continuous  figure,  the  perspective  of  the  further 
edge  of  each  segment,  as  seen  from  the  lower  station- 
point,  coinciding  with  that  of  the  hither  edge  of  the 
segment  below,  as  seen  from  the  station-point  next 
above. 

372.  Alongside,  to  the  right,  drawn  to  the  same  scale, 
is  the  perspective  plan  of  the  circle.  The  left  side  shows 
the  perspective  of  its  projection  upon  the  ground-plane, 
a  g.  This  is  an  ellipse,  divided  into  the  four  segments  cor- 
responding to  those  into  which  the  plan  is  divided.  The 
right-hand  side  shows  the  perspectives  of  the  projections 
upon  the  three  sunken  jdanes,  wliich  have  their  ground- 
lines  at  (/',  g'",  and  ^^^  as  they  appear  when  viewed, 
respectively,  from  S",  S'",  and  S^"^.  The  second  seg- 
ment of  the  second  ellipse,  the  third  of  the  third,  and 
the  fourth  of  the  fourtli,  are  extended  so  as  to  be  as 
wide  as  the  first  segment  of  tlie  first,  and,  like  it,  are 
drawn  in  with  a  full  line,  thus  making  a  continuous 
figure,  in  which  the  remoter  parts  of  the  circular  surface 
are  as  advantageously  displayed  as  the  nearer  parts. 

373.  In  Fig.  105,  h,  we  see  that  exactly  the  same 
Fig.  105,  b.  result  is  reached  if,  instead  of  thus  using  suc- 
hldined^^  cessive  horizontal  planes,  we  employ  Mr.  Ad- 
p'^^o^s-  hemar's  device  of  successive  inclined  planes. 
Not  only  in  tliis  case  do  the  perspectives  themselves 
coincide  at  all  points,  but  the  ground-lines  and  horizons 
of  the  inclined  planes  are  identical  with  those  of  the 


SUCCESSIVE   INCLINED    TLANES.  243 

horizontal  planes  in  the  previous  figure.  The  practical 
work  of  thus  constructing  the  perspective  drawing  is 
then  the  same,  whichever  method  we  prefer  to  employ. 
In  either,  the  dimensions  of  width  and  Iieight  may  be 
laid  off',  according  to  the  scale  of  the  picture,  upon  the 
successive  ground-lines  and  upon  the  vertical  margins. 
The  dimension  of  depth  in  each  plane  may  also  be  laid 
off  upon  the  same  ground-line,  and  transferred  to  the 
line  of  depths  by  the  successive  points  of  distance, 
which,  upon  each  Horizon,  will  give  the  common  dis- 
tance of  the  successive  station-points  from  the  picture. 

374  This  is  illustrated  in  tlie  right-hand  side  of 
Fig.  105,  d,  which  shows  the  successive  per- 
spective plans  figured  in  Fig.  105,  h,  upon  a 
scale  double  that  of  the  previous  figures.  In  two  of 
the  four  segments  shown  in  plan  in  Fig.  105,  a,  a  point 
is  taken  at  random,  and  its  co-ordinates,  measured  from 
the  ground-line  and  from  a  line  of  depths  erected  at  L, 
are  shown.  These  points,  numbered  1  and  4,  in  the  first 
and  fourth  planes,  are  put  into  perspective  by  means  of 
the  successive  ground-lines  gl  and  g^^  V-^,  and  the  points 
of  half-distance,  D^  and  D^^"^.  Any  number  of  points 
of  the  circle,  or  of  any  other  figure,  could  be  put  into 
perspective  upon  the  successive  planes  in  the  same  w^ay, 
just  as  in  Fig.  92,  Plate  XX.  (330). 

375.   These  figures  show  also  that  if,  in  either  fio-ure, 
105,  h,  or  c,  we  consider  the  vertical  planes  Tijgj.esuit3 
which  divide  the  plan  into  segments  to  be  so  '^*^°*'*'^^- 
many  successive   planes   of  measures  (93)  intersecting 
the  four  successive  ground-planes,  whether  horizontal 


244  MODERN  perspective: 

or  inclined,  in  the  ground-lines  gj^i,  gj.-^,  ^4/4,  the  per- 
spectives of  the  successive  segments  will  form  a  con- 
secutive series.  Whether  viewed  from  S  in  Fig.  105,  h, 
or  from  the  successive  station-points  of  105,  c,  they  will 
present  the  appearance  of  Fig.  105,  d.  This  is  an  addi- 
tional illustration  of  the  point  made  in  a  preceding 
paragraph  (358),  and  illustrated  in  Figs.  99  and  100  of 
the  previous  plate,  that  when  a  perspective  drawing  is 
enlaroed  to  a  cdven  scale  it  makes  no  difference  in  the 
result  at  what  distance  from  the  station-point  the  plane 
of  measures  in  the  orthographic  plan  is  set,  and  that,  if 
convenient,  part  of  a  picture  may  be  put  in  by  means  of 
one  such  plane,  with  its  corresponding  ground-line,  line 
of  depths,  and  point  of  distance,  and  part  by  means  of 
another. 

376.    This  also  is  illustrated  in  Fi!_^  105,  d,  in  which 
Fig.  105,  rf.     the  methods  of  Figs.  99  and  100  are  applied 

Small-scale 

data.  to  produce  exactly  tlie  same  outline  upon  the 

left-hand  side  of  the  figure  by  the  method  of  small- 
scale  data  as  has  been  produced  on  the  right-hand  side 
by  the  method  of  co-ordinates  (374).  The  figure  shows 
the  successive  ground-lines,  gl,  g^^,  //g^g,  gj^,  and  the 
successive  lines  of  depth,  g  C,  g^y\  //3C",  and  ,^740^^. 
The  successive  origins  of  co-ordinates,  0,  0^,  Og,  and  04, 
are  determined  as  in  Figs.  99  and  100,  and  the  small- 
scale  data  taken  from  the  orthographic  ])lan  are  laid 
off  from  them  by  aid  of  tlie  successive  fractional  points 
of  distance,  Dj,  1)3,  Dg,  and  1)4.  The  co-ordinates  of 
the  points  1,  2,  3,  and  4,  on  the  left-hand  side  of  the 
circle,  corresponding  to  those  already  found  on  the  right- 


SMALL-SCALE   DATA.  245 

hand  side,  are  thus  used  in  the  figure,  and  their  per- 
spectives found. 

377.  This  process,  also,  is  accordingly  equally  avail- 
able, and  is  identical  in  its  methods  and  results,  whether 
we  suppose  the  successive  ground-planes  to  be  inclined, 
as  Mr.  Adheniar  has  them,  or  to  be  horizontal,  as  we  have 
suggested,  and  as  seems  rather  more  in  harmony  with  the 
conceptions  of  this  treatise,  and  with  the  habitual  use  of 
the  sunk  perspective  plan  to  effect  the  same  ends. 

378.  We  have  seen,  in  Pig.  103,  c  (365),  that  if  we 
constantly  lowxr  the  ground-plane  so  as  to  get  ^ 

^  o  J:  o  In  practice 

a  better  view  of   its    more  remote   parts,  we  leed^uorbe 
must  at  the  same  time  raise  the  station-point 
aud  the  Horizon.     Otherwise,  the  perspective  plan  will 
be  dislocated,  as  shown  in  Fig.  101,  d. 

But,  in  practice,  this  stretching  out  of  the  more  remote 
portion  of  a  plan  is  generally  needed  not  in  its  whole 
extent,  but  only  in  part,  often  only  for  particular  objects. 
Such  isolated  details  are  much  more  simply  and  con- 
veniently treated  by  the  use  of  the  ordinary  sunk  hori- 
zontal plan,  without  moving  the  station-point.  The 
advantage  of  this  method  in  getting  such  auxiliary  oper- 
ations out  of  the  way  of  the  picture  has  frequently  been 
illustrated  in  these  pages ;  as,  for  example,  in  Fig.  20, 
Plate  VII. 

379.  Figs.  106, 107, 108,  and  109,  Plate  XXIII,  show 
tlie  application  of  this  method  to  the  interior 

PI  7-r     7  7  7^7  r         •  -r.        •  rT^^  ^'''^^^  XXITI. 

01   the  Halle  aux  Bles,  m  Pans.     The   mam  Figs.  io6, 107, 

lines  of  these  figures  are  taken  from  Mr.  Ad- 

hemar's  treatise,  though  the  construction  lines  are  some- 


246  MODERN   PERSPECTIVE. 

what  differently  adjusted,  and  the  notation  altered.  It 
is  hardly  necessary,  after  what  has  been  said,  to  enter 
into  a  detailed  explanation  of  these  examples.  They 
reproduce  the  main  features  of  Figs.  99  and  100,  as  well 
as  of  Figs.  103  and  105.  Two  points,  a  and  h,  in  the 
second  and  third  planes  respectively,  are  ascertained  by 
the  use  of  small-scale  data. 


CHAPTER  XYI. 

THE  INVERSE  PROCESS. 

380.  It  is  sometimes  desirable  to  invert  the  proced- 
ure  described  in  the  preceding  chapters.  Instead  of 
beginning  with  an  orthographic  plan  and  elevation,  de- 
rivins:  thence  a  perspective  plan  and  perhaps,  Given  the 

^  i.         i  X  perspective 

as  in  the  previous  chapter,  a  perspective  eleva-  J^j^^^^j.^^^^ 
tion,  and  then  finally  arriving  at  a  complete  d^^^^^io^^- 
perspective  drawing,  it  is  often  possible  by  a  reverse 
process  to  derive  the  perspective  plan  and  elevation  from 
the  drawing,  and  from  them  to  obtain  the  actual  shape  of 
the  object,  and  its  relations  to  the  spectator  and  the  plane 
of  the  picture.  Its  dimensions  can  also  be  determined 
if  the  dimensions  of  certain  lines  in  it  are  known. 

To  effect  this  with  any  approach  to  precision  it  is 
necessary  that  the  perspective  lines  shall  make  a  suffi- 
ciently large  angle  with  each  other  or  with  the  Horizon 
clearly  to  indicate  the  position  of  the  vanishing-points ; 
that  is  to  say,  the  object  shown  must  either  be  large, 
near  at  hand,  or  considerably  above  or  below  the  eye. 

381.  If  the  object  is  in  oblique,  or  three-point  per- 
spective, and  its  three  vanishing-points  can  be  obiique,  or 

three-point 

fixed  with  precision,  there  is  no  difficulty,  as  perspective. 
has  been  shown  in  Chapter  VII.,  in  determining  the  posi- 
tion of  the  spectator.    This  fixes  the  centre  of  the  picture, 


248  MODERN   PEKSPECTIVE. 

the  station-point,  the  distance  of  the  station-point  from 
the  centre  and  from  each  of  the  vanishing-points,  and 
all  the  points  of  distance.  For  lines  connecting  the  three 
vanishing-points  represent  the  three  horizons,  and  the 
meeting-point  of  the  perpendiculars  let  fall  from  the 
angles  of  the  triangle  thus  formed  upon  the  opposite 
sides,  is  the  centre,  Y^ ;  the  distance  of  the  station-point 
in  front  of  tliis  point,  and  its  distance  from  each  of  the 
vanishing-points,  is  then  easily  determined  (168),  and 
the  points  of  distance  found.  If,  then,  the  length  of 
any  of  the  vanishing  lines  is  known,  a  line  of  measures 
parallel  to  one  of  the  horizons  can  be  drawn  through 
one  of  its  extremities,  its  true  dimension  according  to 
the  scale  of  the  drawing,  or  of  that  part  of  it,  found  by 
means  of  a  point  of  distance,  and  the  scale  of  the  draw- 
ing and  the  dimensions  of  every  otlier  part  ascertained. 

382.  This  is  illustrated  in  Plate  XXIV.,  Fig.  110.  If 
Plate  XXIV.  we  supposc  the  perspective  of  the  rectangular 
Fig.  no.  i^iock  to  be  given,  the  vanishing-points  Y\ 
V^,  and  V^,  and  the  traces  that  connect  them,  can  be 
obtained,  the  ground-lines,  or  lines  of  measures  o  m,  I  m,, 
and,  if  necessary,  /  o,  drawn,  and  the  relative  dimensions 
of  the  edizes  determined.  Whether  the  l^lock  is  larc^e 
or  small  cannot  of  course  be  learned.  The  size  of  the 
miniature  block,  supposed  to  be  in  contact  with  the 
picture  at  a,  is  determined,  its  edges. being  equal,  to  a  I, 
am,  and  ao;  but  there  is  no  means  of  knowing  how 
much  larger  the  block  itself  is  than  this  miniature  rep- 
resentative. To  determine  tliis  we  must  know  either 
the  actual  dimensions  of  one  edge  of  the  block,  or  the 


THE  INVERSE  PROCESS.  249 

distance  of  the  block  behind  the  picture.     Neither  of 
these  can  be  shown  by  the  picture  itself. 

383.  If  the  object  is  dra\yn  in  two-point,  or  angular 
perspective,  as  is  generally  the  case,  it  does  not  Two-point, 
suffice  for  the  determination  of  its  shape,  posi-  perspective, 
tion,  and  relations  to  the  spectator  that  the  vanishing- 
points  of  its  principal  lines  should  be  known.  For 
fixing  two  vanishing-points  does  not,  as  fixing  three 
does,  determine  the  position  of  the  spectator  and  of  the 
centre  of  the  picture,  and  thence  of  the  points  of  dis- 
tance, nor  does  it  determine  the  attitude  of  the  object, 
or  the  angles  its  sides  make  with  the  plane  of  the 
picture.  Fixing  the  vanisliing-points  only  restricts  the 
locus  of  the  spectator's  position  to  the  semicircle  sub- 
tended by  the  line  joining  them  ;  they  determine  neither 
the  attitude  of  the  object  nor  its  shape.  In  yigin 
Fig.  Ill,  for  instance,  we  have  at  A  and  B  the 
same  perspective  and  the  same  vanishing-points.  But 
at  A  the  station-point,  S,  and  the  centre,  V*^,  are  assumed 
to  be  well  over  towards  the  right,  and  at  B  towards  the 
left.  The  perspective  plans  and  the  elevations  derived 
from  them  are  sliown  below.  The  plans  are  alike,  but 
the  points  of  distance  being  different  tlie  dimensions 
found  upon  the  ground-lines  are  different,  and  tlie  pro- 
portions of  the  building  and  the  slope  of  the  roof  come 
out  differently.  But  while  the  buildings,  though  differ- 
ing in  size  and  shape,  are  alike  in  perspective,  the  doors 
and  windows,  which  are  of  the  same  size  and  shape  in 
one  building  as  in  the  other,  come  out  differently  in 
perspective. 


250  MODERN    PEKSPECTIVE. 

384.  In  order  to  interpret  correctly  a  drawing  made 
The  centre  ^^  angular,  or  two-point  perspective,  it  is 
yc,  given.  ncccssary  to  have  definite  information  as  to  the 
position  either  of  the  centre,  V^,  of  the  vanishing-point 
at  45°,  V-^,  or  of  one  of  the  points  of  distance,  D^ 
or  D^.  The  centre  is  generally  nearly  in  the  middle  of 
the  picture,  but  that  it  is  exactly  there  is  not  to  be 
taken  for  granted.  Its  position  is  often  precisely  indi- 
cated, however,  by  some  secondary  object,  which  is 
drawn  in  parallel  perspective ;  and  it  is  always  a  good 
plan  to  introduce  some  such  object  as  the  pile  of  boards 
in  Fig.  Ill,  A,  as  a  guide  to  the  spectator. 

385.  It  often  happens,  however,  especially  in  archi- 
The  vanish-  tcctural  drawiugs,  that  the  nature  of  the  sub- 
45°  given  to     ject  is  such  as  to  furnish  dia^fonal  lines  lyin^jj 

find  S  and  VC.  ^  "  . 

Fig.  112.  at  45°  with  the  principal  directions,  lines  that 
we  have  called  X  dividing  the  angle  made  by  the  lines 
E  and  L.  Tig.  112  shows,  by  means  of  a  little  ele- 
mentary geometry,  how  in  this  case  the  station-point,  S, 
is  to  be  found,  V^,  V^,  and  V^  being  given.  As  the 
angle  Y^  S  Y^  is  an  inscribed  angle  of  45°,  its  sides 
must  include  an  arc  of  90°.  A  line  drawn  from  x  in 
the  figure  through  V-^  to  the  opposite  circumference 
fixes  the  position  of  S,  and  hence  of  V^,  D^,  and  D^ 

386.  It  is  not  often  that  the  position  of  either  of  the 
Some  real  poiuts  of  distaucc  cau  bc  detected  by  mere 
blhTg^kno^wn,  Inspcctiou  of  the  picture.  But  it  often  hap- 
poin"ofdis-    pens  that  the  real  or  proportional  dimensions 

of  some  of  the  lines  in  the  picture  are  known. 
In  that  case  one  of  the  points  of  distance  can  be  ascer- 


THE  INVERSE  PROCESS.  251 

tained,  and  the  other  elements  of  the  problem  then  easily 
determined. 

Let  us  suppose,  for  instance,  in  Fig.  113,  that  the  rise 
and  tread  of  the  steps  are  known  to  be  six  and  twelve 
inches.  A  line  of  equal  measures,  I  r,  laid  off  parallel 
to  the  Horizon,  from  the  front  edge  of  the  first  step,  in 
length  equal  to  twice  the  vertical  edge,  forms,  with  the 
horizontal  line  in  perspective,  and  a  third  line,  joining 
their  further  ends,  the  three  sides  of  an  isosceles  tri- 
angle. The  vanishing-point  of  the  third  line,  -^6  point 
the  base  of  the  triangle,  gives  the  point  of  being*grv?n 
distance,  D^.  The  distance  from  this  point  to  etc.  '  ' 
its  corresponding  vanishing-point  is  the  dis- 
tance of  the  station-point  from  that  vanishing-point; 
that  is  to  say,  D^  V^=V^2.  S,  which  must  lie  somewhere 
in  the  semicircle  of  which  V^  Y^  is  the  diameter,  is 
then  easily  found,  as  in  the  figure.  V^,  D^,  and  V-^, 
immediately  follow. 

387.   When  the  drawing  to  be  interpreted  is  made  in 
parallel  perspective  it  is  generally  easy  enough  one-point, 

.  or  parallel 

to  find  the  centre  of  the  picture,  the  vanish-  perspectire. 
ing-point  of  the  lines  perpendicular  to  it.  But,  as  in 
the  previous  case,  it  is  impossible  to  tell  what  is  the 
real  shape  of  the  objects  represented,  or  to  know  from 
what  distance  the  picture  should  be  looked  at,  unless  the 
real  shape  of  some  one  of  the  objects  is  known  inde- 
pendently. If  Fig.  20,  for  instance,  Plate  VL,  is  looked 
at  from  a  point  about  three  inches  in  front  of  V^,  as  may 
be  done  by  looking  at  it  through  a  pin-hole,  so  as  to 


252  MODERN    PERSPECTIVE. 

obtain  a  clear  image  on  the  retina,  the  little  pavilion 
represented  looks  about  square,  the  steps  on  the  side 
seeming  very  steep.  Seen  from  a  distance  of  several 
feet  it  looks  two  or  three  times  as  long  as  it  is  wide,  and 
the  steps  seem  of  very  easy  grade.  The  posts  at  the 
corners  are  presumably  square,  and  the  lines  of  the 
pavement  and  of  the  hips  of  the  roof,  in  plan,  are  pre- 
sumably directed  to  the  vanishing-point  of  45°,  which 
is  the  point  of  distance ;  and  the  steps  have  presumably 
the  same  slope  as  those  on  the  right.  The  point  of 
distance  can  be  found  by  the  same  method  as  in  the 
preceding  paragraph,  and  the  true  shape  of  all  the 
objects  in  the  picture  determined. 

388.  Of  course  these  results  are  based  upon  the 
understanding  that  the  objects  represented  are  rectan- 
Acute  and      gular.     If  tlic  liucs  that  define  them  are  known 

obtuse  angles. 

Fig  114  to  form  acute  or  obtuse  angles,  instead  of  an- 
gles of  90°,  the  line  joining  their  vanishing-points  must 
be  treated  as  the  chord  of  a  circle  instead  of  as  a  diame- 
ter, as  is  done  in  Fig.  114. 

At  A  is  shown  an  obelisk  in  perspective,  presumably 
square  in  plan.  The  methods  described  in 
'  the  previous  paragraphs  suffice  to  determine 
successively  the  principal  vanishing-points,  Y^,  Y^,  and 
Y^"^,  the  centre,  Y^,  the  points  of  distance,  D^  and  D^ 
and  a  perspective  plan.  The  dimensions  can  then  be 
determined,  according  to  the  scale  of  the  drawing,  and 
that  scale  may  be  determined  if  any  one  of  the  dimen- 
sions is  known. 


THE  INVERSE  PROCESS.  253 

At  Fig.  114,  B,  is  another  drawing,  the  horizontal  and 
vertical  lines  of  which  are  identical  with  the 

Fig.  114,  B. 

first.     But  this   obelisk  is  understood  to  be 

triangular  and  equilateral,  with  angles  of  60°  instead 

of  90°.     A  perspective  plan,  with  the  vanishing-points 

Y^  and  V^,  are  easily  determined,  as  is  also  V-^,  the 

vanishing-point  of  the  line  bisecting  the  solid  angle  in 

contact  witli  the  picture.      These   elements   suffice  to 

determine  the  orthographic  plan.  Fig.  114,  C.     As  the 

angle  at  the  station-point,  S,  is  only  60°,  in  place  of  90°, 

it  is  included  in  an  arc  of  240°,  the  point  S 

.      .       Fig-  114,  c. 
lying  somewhere   in   that   arc,   which    is   its 

locus.  The  point  Y-^,  however,  enables  us,  as  the  point 
Y-'^'  did  in  the  previous  case,  to  fix  the  exact  position  of 
S,  by  drawing  a  line  through  the  summit  of  the  arc  at 
X,  and  the  point  \^.  If  then  the  eye  is  placed  in 
front  of  Fig.  114,  A,  opposite  Y^',  at  the  distance  indi- 
cated by  S',  on  the  plan  below,  the  obelisk  will  look 
square ;  if  it  is  placed  opposite  Y^,  in  Fig.  114,  B,  at  the 
distance  indicated  by  S,  it  will  appear  to  have  the 
section  of  an  equilateral  triangle. 

389.  The  little  pyramid  on  top  is,  however,  differ- 
ently  drawn  in  the  two  cases,  and  the  position  of  the 
apex  suffices  to  show  that  the  upper  figure  has  four 
sides,  the  lower  but  three.  The  angles  at  which  these 
sides  meet,  however,  is  necessarily  intermediate. 

390.  The  fact  that  acute  or  obtuse  angles  can  thus  be 
interpreted  as  right  angles  makes  it  difficult  to  repre- 
sent them  satisfactorily  when  there  is  nothing  else  in 
the  picture  to  guide  the  judgment.     It  often  hai)pens  in 


254  MODEKN  PERSPECTIVE. 

the  case  of  buildings  situated  where  two  streets  meet  at 
an  odd  angle  that  drawings  of  them  look  as  if  the  build- 
ings were  square.  To  obviate  this  it  is  necessary,  as 
has  been  said,  to  introduce  something  which  is  unmis- 
takably rectangular,  such  as  an  awning,  or  chimney,  or 
a  cart  backing  up  to  the  sidewalk,  like  the  pile  of  boards 
in  Fig.  Ill,  A. 

391.  If  it  were  not,  indeed,  for  the  facility  with  which 
the  mind  thus  gives  the  most  reasonable  interpretation  to 
the  phenomena  that  meet  the  eye,  even  the  right  angles 
shown  in  perspective  would  generally  look  either  acute 
or  obtuse,  since  it  is  only  when  the  eye  is  exactly  at  the 
station-point,  or  rather  when  it  is  on  the  circumference 
of  a  semicircle  lying  between  the  vanishing-points,  that 
the  angle  really  looks  as  a  right  angle  would.  But  these 
distortions,  like  the  other  distortions  that  arise  from 
abandoning  the  station-point,  are  made  light  of  by  the 
intelligence.  It  is  only  in  the  case  of  circles,  cylinders, 
and  spheres  that  one  is  disturbed  by  them.  In  those 
cases,  indeed,  remaining  at  the  station-point  hardly 
suffices  to  reconcile  one  to  the  drawing,  as  has  been 
explained  in  Chapter  XII. 


CHAPTEE  XYII. 

SUMMARY.  —  PEIXCIPLES. 

392.  Let  us  now  review  the  groimcl  gone  over  in 
the  previous  chapters,  giving  the  subject  a  somewhat 
more  formal  treatment.  We  can  then  proceed  to  ex- 
tend the  principles  and  metliods  employed  to  the 
solution  of  some  problems  which  we  have  not  as  yet 
encountered, 

Projwsitions  and  Definitions. 

393.  A  perspective  drawing  is  a  representation  of  the 
lines  that  would  result  from  the  projection  upon  a  trans- 
parent plane  of  a  plane  or  solid  figure  situated  behind 
that  plane,  the  lines  of  projection  being  visual  rays  con- 
verging from  every  point  of  the  figure  to  the  eye  of  a 
spectator  situated  in  front  of  the  plane. 

This  sort  of  projection  is  called  Conical  Projection,  as 
distinguished   from    Orthographic    Projection,  pj^^^  ^^^ 
in  which  the  lines  of  projection  are  parallel,  ^^^'  ^^^' 
and  are  normal  to  the  plane  of  projection.    See  Fig.  115, 
Plate  XXV. 

The  transparent  plane  of  projection  is  called  .^^^  perspec- 
the  Perspective  Plane.  '"'  p'"^"" 


256  MODERN    PERSPECTIVE. 

The  position  of  the  spectator  is  called  the  Station- 

The  station-      poillt. 

point,  centre,  .  .         ,  .  , 

and  axis.  1  he  pomt  111  the  perspective-plane  opposite 

the  station-point  is  called  the  Centre.  It  is  the  ortho- 
grapliic  projection  of  the  station-point  upon  the  per- 
spective-plane, and  is  the  point  in  that  plane  nearest 
the  spectator's  eve.  It  is  not  necessarily  in  the  middle 
of  the  picture. 

The  line  of  projection  drawn  from  the  station-point 
normal  to  the  perspective-plane  and  passing  through 
the  Centre,  is  called  the  Axis.  Its  length  gives  the  dis- 
tance of  the  eye  in  front  of  the  perspective-plane. 

Unless  otherwise  expressly  mentioned,  the  perspective- 
plane  is  understood  to  be  vertical,  the  Axis,  accordingly, 
horizontal,  and  the  Centre  on  a  level  with  the  eye  at  the 
station-point.  The  eye  is  supposed  always  to  remain  at 
the  station-point. 

394.  The  conical  projection  of  a  figure,  line,  or  point 
The  perspec-  upou  the  perspectivc-plane  is  called  its  per- 

tive  repre- 
sentation,      sj^ective.       When    view^ed    from    the    station- 
point  the  perspective  exactly  coincides  with  and  covers 
the  object  itself 

An  object  and  its  perspective  are,  accordingly,  not  to 
be  distinguished  in  the  picture ;  but  it  is  necessary  to 
distinguish  between  them  in  idea,  and  sometimes,  to  pre- 
vent confusion  to  speak  of  the  real  horizon. 

Objects  lying  at  the  same  distance  as  the  perspective- 
pcnie  depend-  plane  are  represented  as  of  their  real   size. 

ent  on  posi- 

tir'tir''lr''*'  ^^^  \\nQ^  and  figures  lying  in  the  perspective 
speetive-piane.  pia^i^Q  are  thclr  own  perspectives. 


PROPOIITIONS   AND    DEFINITIONS.  2o/ 

Objects  behind  the  perspective-plane  have  their  per- 
spectives smaller  than  themselves.  Olijects  in  front  of 
the  perspective-plane  may  be  conically  projected  back 
upon  it,  and  have  their  perspectives  larger  than  them- 
selves. All  objects  have  their  remoter  parts  drawn  to 
a  constantly  smaller  scale  than  their  nearer  parts. 

395.  The  surface  upon  which  the  drawing  is  made  is 
called  the  plane  of  the  paper,  or  plane  of  the  The  plane  of 

^  1.     X  J.  ^^^  picture. 

picture.     See  Fig.  116,  A.  Fig.iie. 

When  the  objects  to  be  drawn  are  very  small,  or  the 
picture  very  large,  the  plane  of  the  picture  and  the  per- 
spective-plane may  coincide,  as  in  Fig.  115.     . 

But  generally  the  objects  to  be  drawn  are  many  feet  in 
dimension,  as  at  B,  Fig.  116,  while  the  drawing  is  to  be 
measured  by  inches ;  and  as  it  is  always  convenient,  for 
practical  reasons,  to  have  the  plane  of  projection  as  near 
tlie  object  as  possible,  and  consequently  of  about  the  same 
size,  while  the  picture  must  be  small  and  near  at  hand, 
tlie  plane  of  the  picture  and  the  perspective-plane  are 
generally  nearly  as  far  from  eacli  other  as  the  spectator 
is  from  the  object,  as  in  Fig.  116. 

In  this  case  the  drawing  is  a  miniature,  or  small-scale 
representation,  of  the  perspective  lines  supposed  to  be 
traced  upon  the  perspective-plane.  It  may  be  regarded 
as  a  perspective  of  the  perspective-plane,  as  if  made 
upon  a  second  transparent  plane,  near  the  eye,  parallel  to 
the  perspective-plane.  All  the  lines  in  the  per- 
spective-plane will  be  represented  unchanged  in 
direction,  and  reduced  in  a  uniform  ratio.     The  scale  of 

17 


258  MODERN   PERSPECTIVE. 

the  drawing  will  depend  upon  the  relative  distance  of 
the  two  planes  from  the  eye  (94). 

396.  The  same  result  may  be  reached  by  conceiving 
the  drawing  to  be  made  not  from  the  object  itself,  many 
feet  distant  and  many  feet  in  dimension,  but  from  a 
miniature,  or  model,  close  to  the  plane  of  the  picture,  or 
even  in  contact  with  it.  Any  line  that  lies  in  the  plane 
of  the  picture  is  then  its  own  perspective  (394).  See 
Fig.  117. 

In  this  case  the  perspective-plane  and  the  plane  of 
the  picture  coincide,  as  in  the  case  of  small  objects.  As 
this  is  the  most  convenient  working  hypothesis  it  is 
generally  adopted,  and  the  distinction  between  the  two 
planes  is  not  recognized. 

In  the  previous  chapters  the  perspective-plane,  where 
it  has  been  necessary  to  speak  of  it,  has  been  called 
the  plane  of  measures  (93).  In  this  chapter  and  the 
following  one  the  perspective-plane  and  the  plane  of  the 
picture  will,  as  usual,  be  considered  identical. 

Planes. 

397.  Every  plane  figure,  whether  horizontal,  ver- 
^.  .,     ^      tical,  or  inclined,  is  resfarded  as  lyinoj  in  and 

Finite  and  '  '  o  JO 

planes^  fomiing  a  part  of  an  indefinitely  extended  or 
Fig.  118.  infinite  plane,  which  in  its  hither,  or  nearer, 
part  intersects  the  perspective-plane  in  a  line  called 
its  initial-line  or  trace.  In  its  further  part  it  tends 
to  reach,  but  can  never  pass,  an  infinitely  distant 
line,  which  is  a  great  circle  of  the  infinite  sphere 
of   which   the   station-point  is  the   centre,  and   which 


PLANES.  259 

is  called  the  vanisbing-line,  or  horizon,  of  the  plane. 
See  Fig.  118. 

The  perspective  of  such  an  infinite  plane  extends, 
in  the  perspective  plane,  from  the  initial-line,  The  initial 
or  trace,  which  is  its  own  perspective,  since  it  aud'hod"^^^^' 
lies  in  the  perspective-plane  (394),  to  the  per-  ^*^"' 
spective  of  its  horizon,  but  not  beyond  it.  The  per- 
spective of  the  horizon  of  a  plane  is  also  called  the 
horizon  of  the  plane. 

398.  Planes  that  are  parallel  to  one  another  are  said 
to  belong  to  the  same  system  of  planes.    They  Parauei 

planes. 

have  the  same  horizon,  which  is  called  the  ^'g-  n^. 
horizon  of  the  system,  but  not  the  same  traces.  The 
initial-lines,  or  traces,  of  parallel  planes,  which  are  the 
lines  in  which  they  cut  the  plane  of  the  picture,  are 
naturally  parallel  to  one  another.  A  plane  drawn 
through  the  station-point,  in  front  of  tlie  picture,  and 
through  the  horizon  of  the  system,  is  a  member  of  the  sys- 
tem seen  edgewise.  It  intersects  the  perspective-plane 
in  the  horizon  of  the  system.  The  initial-line,  or  trace,  of 
this  plane  and  the  perspective  of  its  horizon  coincide. 
See  Fig.  119.  Such  a  plane  is  called  an  Optical  plane. 
The  trace  of  a  plane  is  accordingly  parallel  to  its 
horizon. 

399.  Hence  the  line  in  which  a  plane  passing  through 
the  station-point,  and  parallel  to  a  given  plane,  intersects 
the  perspective-plane,  is  the  horizon  of  the  given  plane 
and  of  the  system  of  planes  to  which  it  belongs.  Such 
a  plane,  since  it  passes  through  the  eye,  is  seen  as  a  line, 
covering  and  coinciding  with  its  horizon  and  the  per- 


260  MODERN    PERSPECTIVE. 

spective  of  its  horizon.  Both  the  horizon  of  a  plane 
and  the  perspective  of  this  horizon  may  accordingly  be 
found  by  looking  in  directions  parallel  to  tliis  plane. 

400.  Hence,  whatever  angle  two  planes,  or  two  sys- 
The  angle  of   tcnis  of  planes,  make  with  each  other  in  space, 

two  plaues. 

Fig.  120.  the  Optical  planes  drawn  through  the  station- 
point  to  their  horizons,  or  to  the  lines  which  are  the 
perspective  of  their  horizons,  will  make  the  same  angle 
at  the  station-point,  in  the  air  in  front  of  the  picture, 
as  do  the  planes  themselves.     See  Fig.  120. 

401.  A  right  line  lying  in  a  plane  is  called  an  ele- 
ment of  that  plane. 

A  right  line  normal  or  perpendicular  to  a  plane  is 
called  an  axis  of  that  plane  or  system  of  planes. 

All  planes  have  an  element,  or  system  of  elements, 
parallel  to  the  picture. 

All  planes  have  an  element,  or  system  of  elements, 
that  is  horizontal. 

402.  Planes  parallel  to  the  picture  have  all  their 
rueuothe  elements  parallel  to  the  picture.  See  Fig.  121. 
fJ^YIi.  The  axes  of  such  planes  are  normal  to  the 
picture  and  parallel  to  the  Axis.  The  traces  of  such 
]>lanes,  and  their  horizons,  are  at  an  infinite  distance, 
and  cannot  accordingly  be  represented.  Lines  and 
figures  in  any  such  plane  are  drawn  at  a  uniform  scale. 

All  the  planes  of  a  system  not  parallel  to  the  picture, 
paSerto  whether  horizontal,  vertical,  or  oblique,  have 
Fig.^iig?'^^'    the  same   horizon  but  not  the  same  traces. 


PLANES.  261 

Their  initial-lines,  or  traces,  are  parallel  to  each  other  and 
to  the  horizon  of  the  system,  as  above.    See  Fig.  119. 

Planes  that  have  elements  that  are  perpendicular  or 
normal  to  the  picture  are  called  normal  planes.  Normal 

,      .  n    1  plaues. 

Their  axes  are  parallel  to  the  picture.  The  ^^s- 122. 
horizons  of  such  planes  pass  through  the  centre  of  the 
picture.  Both  the  horizons  and  their  traces  are  per- 
pendicular to  tlieir  axes,  and  give  the  true  slope  and 
direction  of  the  steepest  lines  of  such  planes ;  that  is 
to  say,  of  the  planes  themselves.  These  steepest  lines 
are  parallel  to  the  picture.     See  Fig.  122. 

Tiie  horizontal  elements  of  normal  planes  are  normal 
to  the  picture. 

403.  Horizontal  planes  are  normal  planes  in  which  all 
the  elements  are  horizontal.  Their  axes  are  vertical,  and 
their  horizon  and  traces  are  horizontal.     The  Horizontal 

plaues. 

horizon  of  horizontal  planes  is  called  the  Hori-  Fig-  us. 
zon.     The  most  important  horizontal  plane  is  called  the 
ground-plane,  and  its  trace  the  ground-line.    See  Fig.  115. 
Other  planes  besides  horizontal  planes  have  only  one 
set  of  elements  horizontal.     If  this  is  parallel  inclined 

J  lanes. 

to  the  picture,  their  traces  and  horizons  will  be  ^^s- 123. 
horizontal ;   i.  e.,  parallel   to  the  Horizon.      These  are 
sometimes  called  inclined  planes.    Their  axes  are  inclined 
also,  and  lie  in  normal  vertical  planes.     See  Fig.  123. 

Other  planes,  besides  those  parallel  to  the  picture, 
have  only  one  set  of  elements  parallel  to  the   vertical 
picture.      If  this  is  vertical  they  are  called   ^^^°^^* 
vertical  planes.      Their   traces   and   horizons   are  ver- 


262  MODERN   PERSPECTIVE. 

tical,  and  their  axes  are  horizontal,  as  shown  at  A,  in 
Fig.  122. 

404.  In  general,  those  elements  of  planes  that  are 
Oblique         parallel  to  the  picture  are  neither  vertical  nor 

planes.  ^ 

Fig.  124.  horizontal,  and  their  horizontal  elements  are 
neither  parallel  nor  normal  to  the  picture.  Such  planes 
are  called  oblique  planes.  Their  traces  and  horizons 
make  an  angle  with  the  Horizon  and  the  ground-line, 
but  this  angle  is  less  than  their  true  slope.  Their  axes 
also  are  oblique,  being  inclined  both  to  the  picture  and 
to  the  ground-plane.     See  Fig.  124. 

Lines. 

405.  Every  line  is  regarded  as  lying  in  and  forming 
Finite  and      part   of  au  indefinitely  extended  or  infinite 

infinite  lines. 

Fig.  125.  line,  which,  if  not  parallel  to  the  perspective- 
plane,  or  plane  of  the  picture,  pierces  it  at  its  hither,  or 
nearer  end,  at  a  point  called  its  initial-point,  and  at  its  fur- 
ther end  tends  to  reach,  but  can  never  pass,  an  infinitely 
distant  point,  called  its  vanishing-point.  See  Fig.  125. 
The  perspective  of  such  an  indefinite  line  is  drawn 

The  initial-     iu  thc  plane  of  the  picture  from  the  initial- 
point,  vanish- 
ing-point, and  point,  which  is  its  own  perspective,  towards, 

its  perspec-        ^  ^         ^ 

tive.  Qr  even  up  to,  the  perspective  of  its  vanish- 

ing-point, but  it  can  never  pass  beyond  this  point.  It  is 
the  line  in  which  the  plane  of  rays,  drawn  from  the  given 
line  to  the  station-point,  intersects  the  perspective-plane. 

406.  Lines  that  are  parallel  to  one  another,  whether 
Parallel  lines,  horizontal.  Vertical,  or  inclined,  are  said  to 
Pig.  126.        belong  to  the  same   system  of  lines.     They 


LINES.  263 

have  the  same  vanishing-point,  which  is  called  the  van- 
ishing-point of  the  system.  A  line  from  the  station- 
point  to  this  vanishing-point  is  a  member  of  the  system, 
seen  endwise.  It  pierces  the  perspective-plane  in  the 
perspective  of  the  vanishing-point.  Its  initial-point 
and  the  perspective  of  its  vanishing-point  coincide.  See 
Fig.  126.  The  portion  of  this  line  lying  in  front  of  the 
picture,  between  the  station -point  and  the  perspective 
of  the  vanishing-point,  is  called  an  Optical  line. 

407.  Hence,  the  point  at  which  a  line  drawn  through 
the  station-point,  parallel  to  a  given  line,  pierces  the 
perspective-plane,  is  the  perspective  of  the  vanishing- 
point  of  the  given  line,  and  of  the  system  to  which  the 
line  belongs.  Such  a  line,  since  it  passes  through  the 
eye,  appears  as  a  point,  covering  and  coinciding  with 
the  vanishing-point  and  its  perspective.  The  vanishing- 
point  of  a  given  line,  and  the  perspective  of  this  vanish- 
ing-point, may  accordingly  be  found  by  looking  in  a 
direction  parallel  to  that  of  the  line. 

408.  Hence,  whatever  angle  two  lines  make  with  one 
another  in  space,  the  optical  lines  drawn  from  ^^^^  ^^  j^ 
the  station-point,  in  front  of  the  picture,  to  the  Sit' ^^  ^""'^ 
perspective  of  their  vanishing-points,  will  make  ^'^'  ^^^' 
the  same  angle  in  the  air  at  the  station-point.   See  Fig.  127. 

409.  Lines  belonging  to  a  system  that  is  parallel  to  the 
perspective-plane,  whether  vertical,  horizontal,  Lines  parai- 

T         T     1  1      •  •  111  lei  to  the 

or  mclmed,  have  their  perspectives  parallel  to  picture, 
themselves  and  to  each  other,  their  initial-points  and  the 
perspectives  of  their  vanishing-point  being  alike  at  an 
infinite  distance  upon  the  perspective-plane.     They  are 
drawn  to  the  same  scale  in  all  their  parts.    See  Fig.  121. 


264  MODERN    PERSPECTIVE. 

Such  are  the  axes  of  normal  planes.  If  such  planes 
are  horizontal  their  axes  are  vertical,  and  have  their 
vanishing-points  in  the  zenith  and  nadir;  if  vertical, 
their  axes  are  horizontal ;  if  inclined,  their  axes  are 
inclined,  in  a  contrary  sense.     See  Fig.  122. 

410.  The  perspectives  of  a  system  of  lines  not  parallel 
Lines  not       to  tlic  perspectivc-plane,  whether  horizontal  or 

parallel  to 

the  picture,  inclined,  are  not  parallel  to  the  lines  them- 
selves nor  to  each  other,  but  converge  towards  the 
perspective  of  their  vanisliing-point.  They  are  drawn 
to  a  constantly  diminishing  scale  in  their  remoter  parts. 
See  Fig.  126. 

Such  are  the  axes  of  oblique  planes,  which  are  oblique 
also,  being  inclined  both  to  the  picture  and  to  the 
ground  plane.  Such,  also,  are  the  axes  of  inclined 
planes  which  have  their  horizontal  element  parallel  to 
the  picture.  Their  axes  also  are  inclined,  in  the  con- 
trary sense,  and  have  their  vanishing-points  in  a  verti- 
cal line  passing  through  the  Centre.  The  axes  of 
all  vertical  planes  have  their  vanishing-points  in  the 
Horizon. 

411.  Lines  normal  to  the  perspective-plane,  and  par- 
Normai  lines.  'T-^lel  to  tlic  axis,  havc  the  centre  of  the  picture 
Fig.  128.        fQj,  ^i^g-j.  vanishing-point.     See  Fig.  128. 

Such  are  the  axes  of  planes  parallel  to  tlie  picture. 
See  Fig.  121. 

Lines  that  lie  in  the  perspective-plane  are  their 
Lines  in  the  owu  pcrspcctivcs  (394),  and  are  consequently 
picture.  drawu  of  their  own  size  and  shape.  See 
Fig.  117. 


LINES.  265 

412.  All  lines  lying  in  or  parallel  to  a  plane 
have  their  vanishino-points  in  the  horizon  of  Lines  in 

""    _  planes. 

the  plane,  and   tlie  lines  that  lie  in  it  have  Fig.  129. 
their    initial-points    in  its   initial-line,   or   trace.     See 
Fig.  129. 

.  [All  the  elements  of  the  ground-plane  have  their 
vanishing-points  in  the  Horizon  and  their  initial-points 
in  the  ground-line.     See  Fig.  115.] 

Converselij :  —  The  horizon  of  a  plane,  passes  through 
the  vanishing-points  of  all  the  lines  that  lie  in  or  are 
parallel  to  it,  and  the  initial-line  or  trace  through  the 
initial-points  of  the  lines  that  lie  in  it. 

[Since  two  lines  are  sufficient  to  determine  a 
plane,  this  proposition  takes,  in  practice,  the  following 
form :  — 

The  horizon  of  a  plane,  passes  through  the  vanishing- 
points  of  any  two  lines  that  lie  in  it,  and  its  initial  line 
or  trace  through  their  initial-points.] 

Hence:  —  The  line  of  intersection  of  two  planes  has 
its  initial-point  at  the   intersection  of  their  a  line  in  two 

■^  planes. 

traces,  and  its  vanishing-point  at  the  inter-  Fig.  130. 
section  of  their  horizons.     See  Fi^.  130.. 

Conversely :  —  The  horizons  of  all  the  planes  that  can 
be  passed  through  or  parallel  to  a  given  line  will  inter- 
sect each  other  at  its  vanishing-point,  and  the  traces  of 
all  planes  passed  through  it  will  intersect  each  other  at 
its  initial-point. 

413.  Since  a  plane  may  be  conceived  to  be  passed 
through  a  line  in  any  direction,  and  its  trace  and  its 


266  MODERN    PEUSPECTIVE. 

horizon  may  accordingly  pass  through  its  initial-point 
and  its  vanishing-point  in  any  direction,  it  follows  that 
any  two  parallel  lines  in  the  perspective-plane,  drawn 
Accidental     at  random  through  the   initial-point  and  the 

traces  and  •    i   •  •     ,       i>  •  ^-  ^        ,     i 

initial-lines,  vauislung-point  01  a  given  line,  may  be  taken 
to  represent  the  initial-line,  or  trace,  and  the  vanishing- 
line,  or  horizon,  of  a  plane  passing  through  the  given 
line. 

Hence  :  —  Any  line  in  a  drawing  that  happens  to  pass 
through  the  initial-point  of  a  perspective  line  may  be 
taken  as  the  trace  of  a  plane  containing  the  perspective 
line,  a  plane  of  wliich  the  liorizon  may  be  found  by 
drawing  a  parallel  line  tlirough  tlie  vanishing-point  of 
the  perspective  line ;  conversely,  any  line  that  passes 
through  the  vanishing-point  may  be  taken  as  the 
horizon  of  a  plane,  and  the  corresponding  trace 
may  be  drawn  parallel  to  it  througli  the  initial- 
point. 

414.  The  perspective  of  the  horizon,  and  the  trace, 
Lines  parallel  or  initial-line,  of  a  plane  are,  as  has  been 
of  a  plane,  showu,  always  parallel  to  each  other,  and  to 
the  initial-lines  of  all  the  other  planes  of  the  system 
(398). 

They  are  parallel  also  to  the  lines  in  which  the  plane 
in  question,  and  all  the  other  planes  of  the  system, 
intersect  all  planes  parallel  to  the  perspective-plane 
and  to  the  perspectives  of  these  lines.  For  the  lines  in 
which  the  two  systems  of  planes  intersect  are  all  par- 
allel, and  parallel  to  the  picture,  and  their  perspectives 


POINTS.  267 

are  accordingly  parallel  to  themselves  and  to  each  other. 
They  are  parallel  also  to  all  other  lines  lying  in  any 
of  the  planes  of  the  systems,  which  are  parallel  to  the 
perspective-plane  and  to  their  perspectives. 

Points. 

415.  A  given  point  is  regarded  as  lying  in  a  given 
line    at   a   known    distance    from    its  initial-  Points  in 

lines. 

point.     See  Fig.  131.  Fig.  isi. 

In  this  case  the  position  of  the  point  may  be  consid- 
ered to  be  given  by  polar  co-ordinates,  the  initial-point 
being  the  origin  of  co-ordinates,  and  the  direction  being 
given  by  the  vanishing-point. 

When  the  given  line  is  normal  to  the  perspective- 
plane  the  initial-point  is  the  orthograpliic  projection  of 
the  point  upon  that  plane.     In  this  case  the  co-ordinates 

of  a  point. 

position  of  the  point  may  be  considered  to  be  Fig.  132. 
given  by  three  rectangular  co-ordinates,  of  which  the 
co-ordinates  of  the  initial-point,  which  lie  at  right  angles 
to  one  another  in  the  perspective-plane,  are  two,  and  the 
given  line,  lying  at  right  angles  to  them  both,  the 
third.     See  Fig.  132. 

416.  The  perspective  of  a  point  is  the  point  at  which 
a  ray  drawn  from  the  given  point  to  the  eye  pierces  the 
perspective-plane. 

A  point  lying  at  the  intersection  of  two  lines  has 
its   perspective   at   the   intersection   of  their   .     .    . 

^         ^  A  point  in 

perspectives.     See  Fig.  129,  a.  "Z^^ 

A  point  lying  in  the   perspective-plane   is  ^^''^"P'^"^- 
its  own  perspective.     See  Fig.  129,  h. 


268  MODERN    PERSPECTIVE. 

The  Points  of  Distance. 

417.  Tiirough  a  line,  the  perspective  of  which  is  given 
by  its  initial-point  and  the  perspective  of  its  vanishing- 
point,  let  an  anxiliary  plane  be  passed,  its  trace  and 
The  point  of   hoHzon  being  drawn  parallel  to  each  other,  in 

distance. 

Fig.  133.  any  convenient  direction  (413).  See  Fig.  133. 
They  will  pass  through  the  initial-point  and  tlie  perspec- 
tive of  the  vanishing-point  of  the  given  line.  Let  now 
equal  distances  be  assumed  upon  this  initial-line,  as  at  m, 
and  upon  the  given  line  itself,  —  not  its  perspective,  —  as 
at  a,  measuring  from  the  initial-point.  The  lines  thus  set 
off  will  form  the  two  sides  of  an  isosceles  triangle  I  m  o, 
lying  in  the  auxiliary  plane,  behind  the  plane  of  the 
picture,  having  one  of  its  sides,  I  m,  in  the  plane  of  the 
picture.  If  now  this  triangle  be  completed  its  base  will 
be  a  line  also  lying  in  the  auxiliary  plane,  and  the 
vanishing-point  of  this  base  will  lie  in  the  horizon  of 
this  plane.  The  perspective  of  this  vanishing-point  will 
be  found  in  the  horizon  of  the  plane,  as  at  D,  in  the 
figure.  This  vanishing-point,  which  is  also  the  vanish- 
ing-point of  all  lines  drawn  parallel  to  the  base  of  the 
isosceles  triangle,  and  consequently  intercepting  equal 
portions  upon  the  given  line  and  the  initial-line,  is  called 
a  point  of  distance  of  the  given  line. 

418.  The  perspective  of  a  point  of  distance  of  any  line 
is  a  point  hardly  less  important  than  is  the  perspective 
itsperspec-  ^^  ^^^  vanisliing-poiiit.  The  latter  serves  to 
^'^'^*  determine  the  direction  in  which  to  draw  the 
|)erspective  of  a  given  line ;  the  former  serves  to  deter 


POINTS    OF   DISTANCE.  269 

mine  its  length.  For  if,  in  the  plane  of  the  picture,  any 
given  length,  as  I  m,  be  laid  off  upon  the  initial-line  of 
the  auxiliary  plane,  and  a  line  be  drawn  across  the  per- 
spective of  the  given  line  towards  the  perspective  of  the 
-point  of  distance,  this  line  will  be  the  pci- ^he  isosceles 
spective  of  the  base  of  the  isosceles  triangle,  mud  the ^^ 
of  which  the  length,  I  m,  taken  upon  the  initial-  ^^^  "^^" 
line  is  one  side,  and  the  lengtli  intercepted  upon  the 
given  perspective-line  is  the  perspective  of  the  other 
side.  An  indefinite  line,  then,  being  given  by  its  per- 
spective, any  required  length  can  be  determined  upon  it 
by  setting  off  this  length  upon  such  an  initial-line,  or 
trace,  and  transferring  it  to  the  perspective  of  the  given 
line  by  means  of  the  perspective  of  a  point  of  distance. 

419.  The  perspective  of  the  point  of  distance  will  lie 
in  the  horizon  of  the  auxiliary  plane,  as  at  D,  as  will  also 
the  perspective  of  the  vanishing-point,  as  at  Y.  Tliese 
two  points,  which  lie  in  the  plane  of  the  picture,  and 
the  station-point,  S,  which  is  in  the  air  in  ^^^^  isosceles 
front  of  the  picture,  form  the  vertices  of  afronSiUe 
triangle  lying  in  front  of  the  plane  of  the  ^^^  ^^^' 
picture,  of  which  one  side,  V  D,  lies  in  that  plane  and 
the  other  t^vo  meet  at  the  spectator's  eye,  or  the  station- 
point.  One  side  of  this  triangle  lies  in  the  horizon  of  the 
auxiliary  plane,  and  is  accordingly  parallel  to  that  side 
of  the  isosceles  triangle,  behind  the  picture,  which  lies 
in  the  initial-line  of  the  auxiliary  plane;  i.  e.,  VD  is 
parallel  to  I  m;  the  side  that  extends  from  the  station- 
point  to  tlie  perspective  of  the  vanishing-point  of  the 


270  MODEIiN   PERSPECTIVE. 

given  line  is  parallel  to  that  side  of  the  isosceles  tri- 
angle which  lies  in  the  given  line ;  i.e.,SYh  parallel 
to  I  a  ;  and  the  side  that  extends  from  the  station-point 
to  the  perspective  of  the  point  of  distance  is  parallel 
to  tlie  base  of  the  isosceles  triangle ;  i.  e.,  S  D  is  par- 
allel to  ma.  The  triangle  in  front  of  the  plane  of  the 
picture  will  accordingly  also  be  isosceles,  the  homolo- 
gous sides  being  parallel  to  the  sides  of  the  isosceles 
triangle  behind  the  picture,  and  the  perspective  of  the 
point  of  distance  will  be  as  far  from  the  perspective  of  the 
vanishing-point  as  the  perspective  of  the  vanishing-point 
is  from  the  station-point ;  i.  e.,  Y  D  is  equal  to  S  V. 

420.  This  gives  the  following  easy  rule  for  finding  the 
perspective  of  a  point  of  distance  ;  — 

Given  the  perspective  of  a  line  by  its  initial-point 
To  find  a       and  the  perspective  of  its  vanishiug-point, — 

point  of  ,  .  .  . 

distance.  find  thc  distancc  from  the  statiou-pomt  in 
front  of  the  picture  to  the  perspective  of  the  vanish- 
ing-point, and  lay  off  this  distance  from  the  perspec- 
tive of  the  vanishing-point  along  any  line  that  passes 
through  it. 

This  line  will  be  the  horizon  of  a  plane  containing  the 
given  line,  and  the  point  attained  will  be  a  point  of 
distance  of  the  given  line. 

If  now  a  second  line  be  drawn  parallel  to  this  horizon, 
through  the  initial-point  of  the  given  line,  it  will  be  the 
initial-line,  or  trace,  of  this  plane,  and  distances  laid  off 
upon  it  from  the  initial-point  may  be  transferred  to  the 
perspective  of  the  given  line  by  drawing  lines  to  the 
perspective  of  the  point  of  distance.  : 


POINTS   OF   DISTANCE.  271 

421.  When  the  given  line  is  inclined  to  the  Axis, 
the  distance  from  the  station-point  to  the  T^g  instance 
given  vanishing-point  is  the  hypotenuse  of  a  rs^inl^Jiut 
rifrht-anQ:led  triano-le,  of  wliich  the  station-  tion-point. 
point  is  the  vertex,  the  Axis  the  height,  and  ^'^•^^^• 
the  distance  from  the  Centre  to  the  given  vanishing- 
point  the  base.     See  Fig.  134. 

When  the  given  line  is  normal  to  the  plane  of  the 
picture,  having  the  perspective  of  its  vanishing-point 
at  the  Centre,  the  distance  of  this  vanishing- 

,         Fig.  135. 

point   from   the  station-point  is  equal  to  the 

length  of  the  Axis,  and   both  isosceles   triangles   are 

right-angled.     See  Fig.  135. 

422.  Since  the  auxiliary  plane  may  be  taken  in  any 
direction,  its  horizon  may  be  drawn  in  any  direc-  Existing 

•   1  •  lines  avail- 

tion  throuQ,h  the  perspective  oi  the  vanishing- aWe  as  traces 

■■  ^  ^  or  as  bori- 

point.  Any  existing  line  that  happens  to  pass  ^ons. 
through  the  perspective  of  the  vanishing-point  is  accord- 
ingly available  as  such  a  horizon,  a  line  parallel  to  it  being 
drawn  through  tlie  initial-point  as  an  initial-line,  or 
trace.  Conversely,  any  existing  line  that  passes  through 
the  initial-point  may  be  used  as  an  initial-line,  and  a 
horizon  may  be  drawn  through  the  vanishing-point  par- 
allel to  it,  as  has  already  been  said  (413). 

423.  The  locns  of  the  perspective  of  the  point  of  dis- 
tance is  then  a  circle  described  about  the  per-  _    , 

^  The  locus  of 

spective  of  the  vanishing-point  as  a  centre,  d^stJnc?  ^^ 
with  a  radius  equal  to  the  distance  of  this  ^'^-  ^^^• 
centre  from  the  station-point.     Such  a  circle  will  be  cut 
by  any  given  horizon  in  two  points.     Each  horizon  then 


272  MODERN   PERSPECTIVE. 

contains  two  such  points  of  distance,  one  of  which 
makes,  with  the  station-point  and  the  perspective  of 
the  vanishing-point,  an  obtuse-angled  isosceles  triangle, 
and  the  other  an  acute-angled  one.  This  last  is  the  one 
commonly  employed.     See  Fig.  136. 

When  the  given  line  is  normal  to  the  picture,  th^ 
locus  of  the  point  of  distance  is  a  circle  described  about 
the  Centre,  with  a  radius  equal  to  the  Axis.  The  two 
points  of  distance  make,  with  the  station-point  and  the 
perspective  of  the  point  of  distance,  similar  and  equal 
isosceles  triangles,  right-angled  at  the  Centre,  one  of 
which  is,  in  general,  as  convenient  to  make  use  of  as 
the  other.     See  Fig.  135. 

The  lines  S  V  and  S  D  are  optical  lines,  and  make  the 
same  angle  at  the  station-point,  S,  as  do  the  other  lines 
of  the  systems  to  which  they  belong,  the  same  angle  as 
do  the  given  line  and  the  base  of  the  isosceles  triangle 
behind  the  picture  (408). 

NOTE. 

Surveying.  —  In  surveying,  the  true  angle  that  any  visible  hori- 
zontal lines  make  with  one  another  may  be  determined  by  finding 
their  vanishing-points  on  the  Horizon,  and  noting  the  angle  these 
points  subtend  at  the  eye.  This  may  be  done  either  with  a 
theodolite,  or  by  using  a  Plane  Table.     See  page  273,  c. 


CHAPTER   XVIII. 

GEOMETRICAL   PROBLEMS. 

424.  Let  us  now  apply  the  propositions  contained  in 
the  preceding  chapter  to  the  solution  of  the  geometrical 
problems  involved  in  the  cliapters  that  have  gone  before, 
adding  some  problems  of  a  more  purely  speculative  in- 
terest, which,  in  the  ordinary  practice  of  perspective 
drawing,  seldom  present  themselves.  We  shall  find  that 
most  of  the  common  problems  of  Descriptive  Geometry, 
those  at  least  which  concern  the  relations  of  right  lines 
and  planes,  are  easily  solved  in  Perspective. 

425.  The  followiug  table  presents  the  substance  of 
the  principles  rehearsed  in  the  previous  chapter  in  a 
condensed  form,  as  maxims  :  — 

Maxims. 

a.  A  plane  passed  through  the  station-point  parallel  to  a  plane 

cuts  the  plane  of  the  picture  in  the  horizon  of  the  plane 
(399).     Fig.  124.     This  plane  is  an  Optical  plane. 

b.  A   line  passed  through  the   station-point   parallel   to  a  line 

pierces  the  plane  of  the  picture  at  the  vanishing-point  of 
the  hue  (407).     Fig.  126.     This  line  is  an  Optical  line. 
Hence  :  — 

c.  Optical  lines  and  planes,  passed  through  the  station-point  and 

through  the  vanishing-points  and  horizons  of  given  lines  and 
planes,  make  the  same  angles  with  each  other  as  do  the 
given  lines  and  planes  (400,  408).     Figs.  120  and  127. 
18 


274  MODERN  PERSPECTIVE. 

d.  If  a  line  is  normal  to  a  plane,  its  vanishing-point  lies  in  a  line 

drawn  through  the  Centre  at  right  angles  to  the  horizon  of 
the  plane. 
[The  theory  of  projections  teaches  that  if  a  line  is  normal  to  a 
plane  its  projection  upon  a  second  plane  is  at  right  angles  to 
the  line  in  which  the  two  planes  intersect.] 

e.  The  initial-line  or  trace  of  a  plane  is  parallel  to  its  horizon 

(398). 

f.  Parallel  planes  have  the  same  horizon,  and  their  initial-lines, 

or  traces,  are  parallel  to  it  and  to  each  other  (398).    Fig.  119. 

g.  Parallel  lines  have  the  same  vanishing-point  (406).     Fig.  126. 
h.    Lines  lying  in  or  parallel  to  a  plane  have  their  vanishing- 
points  in  its  horizon  (412).     Figs.  115  and  129. 

i.    Lines  lying  in  a  plane  have  their  initial-points  in  its  initial- 
line,  or  trace  (412).     Figs.  115  and  129. 
Conversely :  — 

j.  The  horizon  of  a  plane  passes  through  the  vanishing-points  of 
all  the  lines  (or  of  any  two  lines)  that  lie  in  or  are  parallel 
to  it  (412). 

k.  The  initial-line,  or  trace,  of  a  plane  passes  through  the  initial- 
points  of  all  the  lines  (^.  e.,  of  any  two  lines)  that  lie  in  it 
(412). 

L    A  line  parallel  to  the  plane  of  the  picture  has  its  perspective 
parallel  to  itself  and  to  the  horizons  and  traces  of  all  planes 
that  pass  through  or  are  parallel  to  it  (409).     Fig.  121. 
Conversely :  — 

m.  All  the  planes  that  pass  through  or  are  parallel  to  a  line  that 
is  parallel  to  the  plane  of  the  picture  have  their  traces  and 
horizons  parallel  to  it  and  to  its  perspective  (414). 

n.   The  line  of  intersection  of  two  planes  has  its  initial-point  and 
vanishing-point  at  the  intersection  of  their  traces  and  hori- 
zons (412).     Fig.  130. 
Conversely :  — 

o.  The  traces  of  all  the  planes  that  pass  through  a  line  intersect 
each  other  at  its  initial-point  (412). 


NOTATION.  275 

p.  The  horizons  of  all  the  planes  that  pass  through  or  are  parallel 
to  a  line  intersect  each  other  at  its  vanishing-point  (412). 

q.  The  Centre  is  the  projection  of  the  station-point  upon  the 
plane  of  the  picture  and  the  vanishing-point  of  lines  normal 
to  it  (393,  411).     Fig.  128. 

r.  The  length  of  the  Axis  is  the  distance  of  the  station-point 
from  the  centre  (393).     Fig.  128. 

s.  The  length  of  the  Optical  line  is  the  distance  of  the  station- 
point  from  the  vanishing-point. 

Notation. 
426.  The  following  system  of  Notation  will  be  adopted 
in  this  chapter.     It  is  conformable  to  that  used  in  the 
previous  chapters. 

The  Perspective  Plane PP. 

Tlie  plane  of  the  picture pp. 

The  station-point S. 

The  station-point  when  revolved  into  the  plane  of  the  picture     .     .    .    S',  S",  etc. 

A  line  is  designated  by  a  Roman  capital :  — 

Horizontal  lines,  right  and  left R,  R',  etc.  ;  L,  L',  „ 

Lines  at  45^  (bisecting  a  right  angle) X,  Y  ;  X',  Y',  „ 

Vertical  lines  (to  the  zenith) Z,  Z'  ,, 

Oblique  lines  to  the  right,  up  and  down M,  M',  ,, 

Oblique  lines  to  the  left,  up  and  down N,  N',  ,, 

Lines  normal  to  a  given  plane  (axes  of  the  plane) T,  T',  ,, 

Lines  normal  to  the  plane  of  the  picture C,  C,  ,, 

Oblique  lines  in  normal  vertical  planes A,  A',  ,, 

Lmes  parallel  to  the  plane  of  the  picture,  if  horizontal K,  K',  ,, 

If  inclined  down  to  the  right,  dexter D,  D',  ,, 

If  inclined  down  to  the  left,  sinister S,    S',  ,, 

Sunlight  and  shadow  lines S. 

other  lines E,  J,  O,  U,  etc. 

Optical  lines RO,  LO,  NO,  „ 

Finite  lines  by  small  capitals K,  z,  c,  R,  l,  „ 

The  perspective  of  a  point  is  designated  by  an  italic  letter       .     ,      a,b,  c,  m,  x  „ 
The  vanishing-point  of  a  line  by  V,  with  the  letter  designating  the  line  written 

above  it V^,  V^,  V",  „ 

Other  vanishing-points V,  V,  V"  ,, 


276  MODERN    PERSPECTIVE. 

The  initial  point  of  a  line  by  I,  with  the  letter  designating  the  line  written 
above  it IR,  I^,  IM,  etc. 

Other  initial-points I)  I  »  I"»   >> 

The  points  of  distance  of  a  line  by  D,  with  the  letter  designating  the  line  written 
above  it DR  DSDM,   „ 

Other  points  of  distance D,  D',  D"  „ 

A  plane  is  designated  by  two  Roman  capitals,  indicating  its  horizontal  element 

and  its  steepest  element RM,  LM,  „ 

Horizontal  planes RLj  R'L',  ,, 

Vertical  planes RZ,  LZ,  „ 

Normal  planes CK,  CD,  CS,  CZ,  „ 

Planes  parallel  to  the  picture KZ,  K'Z  „ 

Planes  of  the  shadow  of  lines SR,  SL,  SZ,  „ 

The  horizon  of   a  plane  is  designated  by  the  letter  H,  followed  by  the  letters 

designating  the  plane HRN,  HLM,   „ 

The  Horizon        HRL,  HCK,  or  HH. 

Horizons  of  shadows  of  lines HSR,  HSL,  etc. 

The  initial-line,  or  Trace,  of  a  plane  is  designated  by  the  letter  T,  followed  by 

the  letters  designating  the  plane TRN,  TLM,   „ 

The  Ground-line TRL,  or  gl. 

In  the  plates  a  point  is  shown  thus X 

,,   „      ,,        a  line     „       „  „  

„   „       „       the  trace  of  a  plane  is  shown  thus      .     .     . 

,,   ,,      „       the  horizon  of  a  plane  is  shown  thus  .    .     .    .  .  .  

„  „      „       a  constructive  line        „       „  „         

Data. 

427.  The  elements  that  enter  into  a  perspective  prob- 
lem are,  in  space,  the  relative  positions  of  the  spectator, 
the  perspective-plane,  and  the  plane  of  the  picture,  as 
given  by  the  position  of  the  Centre  and  of  the  Horizon, 
and  the  length  of  the  axis  of  the  planes ;  the  size,  posi- 
tion, and  attit>ide  of  the  object,  as  given  by  the  position, 
magnitude,  and  directions  of  the  lines  and  planes  to  be 
represented ;  and,  in  the  plane  of  the  picture,  the  per- 
spectives of  all  points  and  their  projection  upon  that 


DATA.  277 

plane,  the  perspectives  of  all  finite  lines,  and  the  initial 
and  vanishing-points  of  the  infinite  lines  in  which  they 
lie,  and  the  horizons  and  traces  of  all  planes. 

428.  When,  in  any  given  case,  all  these  things  are 
known,  the  discussion  is  complete.  The  solution  of  a 
problem  in  Perspective  consists  in  showing  how,  where 
only  some  of  these  elements  are  given  or  assumed, 
others  can  be  determined. 

429.  Unless  the  contrary  is  expressly  mentioned,  it 
will  be  understood  that  the  perspective-plane  is  ver- 
tical, and  that  the  position  of  the  Centre,  V^,  that  of 
the  Horizon,  HH,  and  that  of  the  Ground-line,  g  I,  are 
given,  as  is  also  the  length  of  the  Axis,  C^=:S  V^,  which 
determines  the  position  of  the  station-point  in  front  of 
the  picture. 

430.  The  scale  is  generally  supposed  to  be  known ; 
that  is  to  say,  the  dimension  that  shall  be  given,  in  the 
drawing,  to  lines  lying  in  the  perspective  plane.  This 
depends  upon  the  relative  distance  of  the  perspective 
plane  and  the  plane  of  the  picture  from  the  station- 
point  (94),  or  upon  the  scale  adopted  for  the  miniature, 
or  model,  which  is  supposed  to  be  substituted  for  the 
object  itself  (396).  In  that  case  dimensions  can  be 
given  by  figures.  But  in  purely  geometrical  discussions 
they  are  best  given  by  lines. 

431.  A  single  point  may  be  given  by  its  co-ordinates, 
by  its  perspective  in  a  given  line  or  plane,  or  by  its  per- 
spective and  its  projection  upon  a  given  plane.  These 
all  resolve  themselves  into  the  case  of  a  point  lying  in  a 


278  MODERN    PERSPECTIVE. 

given  line.  If  the  point  is  given  by  its  rectangular  co- 
ordinates, or  lies  in  a  plane  normal  to  the  plane  of  the 
picture,  it  lies  in  a  line  normal  to  the  picture;  if  it  lies 
in  an  oblique  plane,  it  lies  in  an  oblique  line;  if  it  is 
given  by  its  projection  on  a  given  plane,  it  lies  in  a  line 
normal  to  the  given  plane. 

432.  Distances  are  measured  from  the  three  plaiies 
passing  through  the  Centre,  as  an  origin  of  rectangular 
co-ordinates,  the  three  axes  of  co-ordinates  being  the 
Axis,  C^=SV^,  normal  to  the  plane  of  the  picture,  and 
passing  tln-ough  the  Centre  and  station-point;  the  Hori- 
zon, HH=HRL,  normal  to  the  vertical  plane  which  pass- 
es through  the  Centre  and  the  station-point  and  which  is 
normal  to  the  plane  of  the  picture ;  and  the  zenith  and 
nadir  line,  ZZ^=HCZ,  normal  to  the  horizontal  plane, 
and,  like  the  Horizon,  lying  in  the  plane  of  the  picture. 
These  three  planes,  namely,  the  Horizontal  plane,  the 
Normal  plane,  and  the  plane  of  the  picture,  are  called 
the  priiicipal  planes. 

I.   Problems  of  Direction. 

These  problems  are  solved  by  the  use  of  orthographic 
projections  upon  the  plane  of  the  picture  and  the  prin- 
cipal horizontal  plane,  according  to  the  common  rules 
of  projection.  Most  of  the  constructions  take  place  in 
front  of  the  plane  of  the  picture,  which  serves  as  the 
vertical  plane  of  projection.     See  Plate  XXVI. 

In  this  plate  the  Horizon,  HH,  the  Zenith  line,  Z2, 
the  Centre,  V^,  and  the  length  of  the  Axis,  C^=SV^,  are 
omitted  when  not  needed  for  the  constructions  shown. 


PROBLEMS    OF    DlUECTION.  279 

Problem  I.  To  find  the  vanishing-point  of  lines 
making  a  given  angle  with  the  horizontal-plane,  their 
projection  upon  that  plane  making  a  given  angle  with 
the  vertical  norn)al  plane  or  with  the  plane  of  the  picture. 

Let  the  lines  M  of  a  given  system  make  the  angle  P 
with  the  horizontal  plane,  and  let  their  projections  upon 
this  plane  make  the  angle  a  with  the  normal  plane. 
Then  the  optical  line,  M^,  belonging  to  this  ^.^^^^  „  ^^^ 
system,  drawn  through  the  station-point,  S,  in  ^*«fi"<^^"- 
front  of  the  picture,  and  having  these  relations  to  the 
principal  planes,  will  pierce  the  plane  of  the  picture  in 
the  common  vanishing-point  of  tlie  lines  of  the  system  (b). 

.-.  Let  li^  be  the  horizontal  projection  of  the  line  M^, 
and  let  it  be  drawn  horizontally  from  the  station-point, 
S,  at  the  angle  a  with  the  Axis,  until  it  touches  the 
Horizon  at  the  point  V^.  This  is  the  vanishing-point 
of  the  horizontal  projections  of  the  lines  M  of  the  given 
system;  that  is  to  say,  of  tlie  lines  parallel  to  E^.  The 
required  vanishing-point,  Y^^,  will  lie  vertically  above 
Y^  in  the  plane  of  the  picture,  and  the  triangle,  Y^'Y^S, 
right-angled  at  Y^,  and  having  V\P  for  its  base  and  M^ 
for  its  hypotenuse,  will  have  the  angle  at  S,  between  ]\I 
and  R,  equal  to  ^  (c).  If  now  tliis  triangle  is  revolved 
about  its  vertical  side,  Y^Y^,  until  it  coincides  with  the 
plane  of  the  picture,  S  will  fall  at  D.  If  the  line  M^  is 
drawn  in  its  revolved  position,  making  at  D  the  angle  y8 
with  the  Horizon,  the  point  Y*^  is  easily  found  at  the 
intersection  of  M^,  in  its  revolved  position,  with  the 
line  erected  vertically  above  Y^. 

Problem  II.   If  the  direction  of  the  given  system  is 


280  MODERN   PERSPECTIVE. 

determined  by  the  angle  <y  which  its  projection  upon  the 
normal  plane  makes  with  the  Axis,  instead  of  by  the 
angle  the  line  itself  makes  with  the  horizontal  plane, 
the  vanishing-point  can  be  found  by  a  process  analogous 
to  the  preceding,  as  in  the  plate. 

Problem  III.  Given  the  direction  of  a  plane,  or  sys- 
tem of  planes,  by  the  direction  of  their  axes,  — 
?ofii?dlt?^'''  To  find  the  horizon  of  the  system  ;  that  is  to 

horizon. 

saij :  — 

Given  a  line  by  its  vanishing-point,  V", — 

To  find  the  horizon  of  planes  normal  to  it. 

.-.  Find  \^,  tlie  vanishing-point  of  the  axes  of  the 
planes  (Prob.  I.  or  II.). 

The  required  horizon  will  be  at  right  angles  to  a  line, 
V^V^  a,  drawn  through  this  vanishing-point  and  the 
Centre,  V^ ;  for  so  much  of  this  line  as  lies  between  V^ 
and  V^  is  tlie  projection  upon  tlie  plane  of  the  picture  of 
a  line  drawn  from  the  station-point  to  the  given  vanish- 
ing-point, V^,  and  the  required  horizon  is  the  line  in  which 
the  plane  of  the  picture  is  intersected  by  an  optical 
plane  drawn  through  the  station-point  normal  to  this  line. 
The  line  of  projection  and  the  line  of  intersection  are 
then  necessarily  at  right  angles  to  one  another  (d). 

It  remains  only  to  find  the  point  a. 

So  much  of  the  line  of  projection  as  lies  between  Y^ 
and  a  is  the  projection  upon  the  plane  of  the  picture 
of  a  line  lying  in  the  optical  plane  at  right  angles  with 
the  first  line.  This  line  passes  through  the  station- 
point  parallel  to  the  given  plane,  and  pierces  the  plane 
of  the  picture  at  the  required  point,  a,  in  the  required 


PROBLEMS    OF  DIRECTION.  281 

horizon.  These  two  lines,  lying  in  the  air  in  front  of 
the  picture,  form  at  the  station-point  the  vertex  of  a 
right-angled  triangle,  right-angled  at  S,  the  liypotenuse 
of  which  is  the  line  of  projection,  V^V^  a. 

If  this  triangle  is  supposed  to  be  revolved  into  the 
plane  of  the  picture  around  this  hypotenuse,  the  Axis 
of  the  picture,  V^S,  will  fall  at  right  angles  to  it  at  V^S'. 
The  point  a,  at  the  otlier  end  of  the  hypotenuse,  is 
tlien  easily  found  by  setting  off  at  S'  the  right  angle, 
A'^S'  a.  The  required  horizon  may  then  be  drawn 
through  a,  at  right  angles  to  Y'^'V*^  a. 

Problem  IV.  Conversely :  Given  the  horizon  of  a 
plane,  or  system  of  planes,  —  Given  a  plane 

To  find  V',  the  vanishing-point  of  lines  nor-  *^i^- 
mal  to  the  system,  —  that  is  to  say,  of  their  axes. 

.*.  This  simply  reverses  the  operations  of  Problem  III. 

If  the  axis  is  parallel  to  the  plane  of  the  picture,  the 
horizon  of  the  system  of  planes  lies  at  right  angles  to  it, 
and  passes  through  the  Centre,  Y^.  If  the  system  of 
planes  is  parallel  to  the  plane  of  the  picture  their  axes 
have  the  Centre,  Y*^,  for  a  vanishing-point. 

Problem  Y.  Given  the  horizon  of  a  plane  and  the 
vanishing-point,  Y,  of  a  line  lying  in  it,  — 

To  find  Y',  the  vanishing-point  of  a  second  line, 
lying  in  the  same  plane,  making  a  given  angle,  (/>,  with 
the  first,  or  perpendicular  to  it. 

Since  the  optical  lines  drawn  from  the  two  vanishing- 
points  to  the  station-point  must  meet  at  the  Given  v 

^  ^  and  4t  to  find 

station-point  in  the  given  angle  (m),  and  lie  ^'' 


282  MODERN    PERSPECTIVE. 

in  a  plane  which  cuts  the  plane  of  the  picture  in  the 
given  horizon  (j), — 

.-.  Eevolve  this  plane  into  the  plane  of  the  picture, 
about  the  given  horizon.  The  projection  of  the  station- 
point  will  move  from  V^  in  a  line  at  right  angles  to  the 
given  horizon,  and  the  station-point  will  fall  in  its  re- 
volved position  at  S",  the  distance,  S"  a,  being  equal  to 
S'  a,  determined  as  in  the  previous  figure,  V^S'  being 
the  length  of  the  Axis. 

If  Y  is  the  given  vanishing-point,  VS"  will  be  the 
revolved  position  of  the  optical  line  of  the  system,  and  a 
line  making  with  it  the  given  angle  <^  will  be  the  optical 
line  of  the  other  system  and  will  cut  the  given  horizon 
at  the  required  vanisliing-point,  Y',  as  in  the  figure. 

If  the  angle  ^  is  90°,  the  two  lines  will  be  at  right 
angles. 

Problem  YI.  Conversely :  Given  the  direction  of 
Given  V  two  liucs,  by  their  vanishing-points,  Y  and 
&id<|).  Y',  to  find   the   angle  they  make   with  one 

another  in  space. 

.-.  This  simply  reverses  the  operations  of  Problem  Y. 

Problem  Y.  a.  If  the  given  line,  lying  in  the  given 

plane,  is  parallel  to  the  plane  of  the  picture, 

line  parallel    g^  ^-[-^^^  ^^^  vauishincr-point,  Y,  is  at  an  infinite 

to  the  pic-  o  i.  '        ' 

*"''®'  distance,  it  will  be  parallel  to  the  horizon  of 

the  plane  (1).  The  line  drawn  through  the  station- 
point  parallel  to  the  given  line  will  accordingly  also  be 
parallel  to  the  given  horizon,  and  may  be  drawn  parallel 
to  it  through  S"  in  its  revolved  position,  as  in  the  figure. 
The  angle  (/>  may  then  be  set  off  as  before. 


PROBLEMS   OF   DIRECTION.  283 

Problem  VI.  a.  Conversely :  By  a  reversal  of  this 
process  we  may  ascertain  the  angle  made  by  a  given  line, 
lying  in  a  given  plane,  with  another  line  lying  in  the 
same  plane  and  parallel  to  the  plane  of  the  picture. 

Problem  VII.  Given  the  direction  of  two  planes  by 
their  horizons  to  find  the  ano^le  between  them.  ^  ^  ^,^ 

o  To  find  the 

[The  angle  made  by  two  planes  is  the  sup-  Jwlen^two 
plement  of  the  angle  made  by  their  axes.]  v^^^^^- 

.*.  Find  the  vanishing-points  of  the  axes  of  the  two 
planes  (Prob.  IV.),  and  then  find  the  angle  made  by 
these  axes.     (Prob.  VI.) 

Problem  VII.  a.  If  the  planes  are  normal  to  the 
plane   of  the   picture,  they   make   the  same  when  the 

planes  are 

angle  with  each  other  as  do  their  horizons.         normal. 

Problem  VII.  b.  If  the  elements  in  the  two  planes 
which  are  parallel  to  the  plane  of  the  picture  when  their 

horizons  are 

are  parallel  to  one  another,  i.  e.,  if  the  horizons  parallel 
of  the  planes  are  parallel,  as  in  Fig.  123,  Plate  XXV., 
the  problem  may  be  solved  by  finding  the  angle  between 
the  planes  themselves  instead  of  that  between  their  axes, 
as  follows :  — 

Find  the  points  a  and  a\  in  the  horizons  of  the  two 
planes,  as  in  Prob.  III.  The  angle  aSa',  revolved  into 
the  plane  of  the  picture  at  aS'a^,  will  obviously  be  the 
required  angle. 

Problem  VIII.  Given  the  direction  of  a  line  by  its 
vanishing-point,  and   that  of  a  plane  by  its  rpofindth 
horizon,  to  find  the  angle  between  them.  tw£A%ne 

[This  angle  is  the  complement  of  that  made  *°^  *  ^^*°®* 
by  the  line  and  the  axis  of  the  plane.] 


284  MODERN   PERSPECTIVE. 

.*.  Find  the  vanishing-points  of  the  axis  of  the  given 
plane  (Prob.  III.),  and  then  find  the  angle  between  this 
axis  and  the  given  line  (Prob.  VII.). 

.'.  Another  way :  Project  the  given  line  upon  the 
given  plane  (Prob.  XXIX.),  and  then  find  the  angle 
between  the  line  and  its  projection  (Prob.  VI.). 

Problem  IX.  Given  the  direction  of  a  line,  or  sys- 
tem of  lines,  M,  by  the  vanishing-point,  V*^,  — 

To  find  (1)  the  optical  line  M^,  i.  c,  the  distance  from 
the  station-point,  S,  to  the  vanishing-point,  V^,  (2)  a 
point  of  distance,  D^,  and  (3)  the  locus  of  the  points  of 
distance  of  the  system. 

(1.)  The  optical  line,  from  the  station-point  to  the 
To  find  a  vanishing-point,  is  the  hypotenuse  of  a  triangle, 
tanceandts  right-angled  at  the  centre  of  the  picture,  V^, 
of  which  the  Axis,  V^S,  is  one  side.  By  re- 
volving this  into  the  plane  of  the  picture  around  its 
base,  V^T^  the  distance,  SV^=M^,  is  easily  found. 

(2.)  This  distance  laid  off,  right  or  left,  upon  the  horizon 
of  any  plane  which  contains  M,  and  which  consequently 
passes  through  V^  (g),  gives  the  points,  D^,  which  are 
called  points  of  distance  of  M.  Their  distance  from  V^ 
shows  the  distance  of  the  station-point  from  V^. 

(3.)  As  the  horizons  of  all  planes  containing  M  pass 
through  V^  (j),  all  possible  points  of  distance  of  the  sys- 
tem M  will  lie  in  a  circle,  of  which  V*^  is  the  centre,  and 
the  distance,  V^^S:=M^,  the  radius.  This  circle  is  the 
locus  of  D^. 

.-.  If  the  given  lines  belong  to  the  system  C,  normal 
to  the  plane  of  the  picture,  tlieir  vanishing-point  is  the 


PROBLEMS   OF   DIMENSION   AND   POSITION.  285 

Centre,  V^,  and  the  radius  of  the  circle,  which  is  the 
locus  of  D^,  is  the  Axis,  V^S.  The  point  of  distance, 
D^,  accordingly,  always  shows  at  once  how  far  the 
station-point  is  from  the  picture.  The  length  of  the 
Axis  is  equal  to  the  distance,  V*^D^=C^. 

The  operations  by  which  the  remaining  problems 
are  solved  are  conducted  in  the  plane  of  the  picture 
by  means  of  vanishing-points,  initial-points,  horizons, 
traces,  and  points  of  distance  which  have  been  previously 
determined. 

II.  Frohlems  of  Dimension  and  Position. 

Problem  X.  To  cut  off  a  given  lenoth,  M,  from  a  line 
given  in  perspective  by  its  initial-point,  I*^,  and  its 
vanishing-point,  V^,  — 

.*.  Draw  through  tlie  initial-point  a  line,  at  random, 
to  represent  the  trace  of  a  plane  passing  ^o  measure 
through  the  line,  and  draw  a  parallel  line  onap'Sspec- 
through  tlie  vanishing-point  to  represent  the 
horizon  of  this  plane  (a,  j,  e).  Find  the  locus  of  the 
points  of  distance  of  the  given  line  (Prob.  IX.).  The 
points  in  which  it  cuts  the  assumed  horizon  will 
be  points  of  distance  of  the  given  line. 

Lay  off  upon  the  initial-line,  or  trace,  at  m,  the  length 
to  be  cut  off  upon  the  line  given  in  perspective,  and 
from  the  point  thus  ascertained  draw  a  line  across  the 
given  line  to  the  opposite  point  of  distance.  The  length 
cut  off  upon  the  given  line  will  be  the  required  per- 
spective length. 


286  MODERN   PERSPECTIVE. 

For,  in  the  figure,  the  line  V^D^^  is,  by  construction, 
parallel  to  I-"^  m,  SV^  (in  the  air  in  front  of  the  picture) 
to  P  a  (b),  and  SD^  (also  in  tlie  air),  to  m  a  (b).  And 
since  the  triangle,  SD^V^  (in  front  of  tlie  picture)  is 
isosceles,  Y^D^  having  been  taken  equal  to  SV^,  the 
triangle  shown  in  perspective  at  V'hn  a  is  isosceles  also, 
the  corresponding  sides  of  the  two  triangles  being  par- 
allel, and  V^a  is  equal  to  I^m. 

It  is  to  be  noticed  that  these  triangles  are  reversed  in 
position. 

Problem  XL  Conversely:  Given  a  finite  line  m  in 
To  find  the  perspective,  or  two  points,  with  the  initial- 
of"ag1vln      point  and  vanishing-point  of  the  infinite  line 

perspective       ,  ,   •    i     ,i  ^^ 

line.  m  which  thcj  lie,  — 

To  find  the  length  of  the  line,  or  the  distance  apart 
of  the  two  points. 

Let  a  and  a'  be  the  given  points,  and  I^  and  Y^  the 
given  initial-point  and  vanishing-point. 

.*.  Through  I*^  and  Y^  draw  parallel  lines  in  any  con- 
venient direction.  These  will  represent  the  trace  and 
horizon  of  a  plane  containing  the  given  points  (e,  j,  k) 
(422).     Find  a  point  of  distance,  D^  (Prob.  IX.). 

From  D^  draw  lines  through  a  and  a',  until  they 
intersect  the  trace  at  m  and  m^ 

The  distance  m  7r)J  is  the  true  distance  of  the  points 
a  a',  or  the  length  M  of  the  line  a  a'. 

If  in  this  and  the  preceding  problem  the  given  points 
or  line  lie  in  a  line  normal  to  the  plane  of  the  picture, 
then  I^,  Y^,  and   D^  take  the  place  of  I^^  Y^,  and 


PROBLEMS    OF   DIMENSION   AND   POSITION.  287 


Problem  X.  a.    To  cut  off  given  lengths  from  lines 


parallel  to  the  plane  of  the  picture.  ^^^^  ^^^^ 

It  is  necessary,  in  order  to  fix  the  position  {J^pTrlneUo* 
of  the  given  line,  that  the  initial-point  and     ^  ^'^ "™' 
vanishing-point  of  an    auxiliary  line   intersecting   the 
given  line  should  also  be  given. 

.'.  Let  ^  be  a  point  in  the  given  line,  p  a,  and  I  and 
V  the  initial-point  and  vanishing-point  of  an  auxili- 
ary line  intersecting  it  at  this  point.  Draw  parallel 
lines  through  V  and  I  to  represent  the  horizon  and 
trace  of  a  plane  containing  both  lines.  They  will  be 
parallel  to  the  given  line  (1),  and  any  distances,  Im, 
mm',  taken  upon  the  initial-line,  may  be  intercepted 
upon  the  given  line  at  7.7  a  and  a  a'  by  lines  parallel  in 
space  (c),  drawn  from  I,  m  and  m',  to  V,  as  in  the 
figure. 

Problem  XL  a.  Conversely :  Given  a  finite  line  par- 
allel to  the  plane  of  the  picture,  with  the  initial-point 
and  vanishing-point  of  an  auxiliary  line  passing  through 
it,— 

To  find  the  real  length  of  the  given  line. 

.'.  This  problem  is  simply  the  reverse  of  the  pre- 
ceding. 


Problem  XTL  To  find  the  perspective  of  a  point 
given  by  its  co-ordinates. 

.*.  Let  the  origin  of  co-ordinates  be  the  Centre,  V^, 
and   let  the  horizontal,  vertical,  and  normal  to  find  the 

perspective 

co-ordinates    of   the    point  be   K,  z,    and    c,  of  a  point. 


respectively. 


288  MODERN   PERSPECTIVE. 

The  normal  co-ordinate  will  lie  in  an  infinitely  long 
line,  C,  passing  through  the  required  point  and  pierc- 
ing the  plane  of  the  picture  at  its  initial-point,  I^. 
Hence :  — 

Lay  off  in  the  plane  of  the  picture,  horizontally  and 
vertically,  from  the  origin,  Y^,  the  co-ordinates  K  and  z. 
They  give  the  position  of  I^,  the  initial-point  of  the 
normal  line  passing  through  the  required  point.  Cut 
off  upon  this  line  the  distance,  c  (Prob.  X.).  The  point 
obtained  will  be  the  perspective  of  the  required  point, 
a,  as  in  the  figure  at  A.  The  point  I^  is  the  ortho- 
graphic projection  of  the  point  a  upon  the  plane  of  the 
picture. 

[The  orthographic  projection  of  a  point  upon  the  plane 
of  the  picture,  the  perspective  of  the  point,  and  the 
centre  of  the  picture,  lie,  of  course,  in  the  same  straight 
line.] 

The  horizon  and  trace  of  the  auxiliary  plane  may 
be  taken  at  random,  as  at  A,  or,  as  is  generally  more 
convenient,  and  as  is  shown  at  B,  may  be  taken  as 
coinciding  in  direction  respectively  with  K  and  the 
Horizon,  HH,  or  with  z,  and  the  zenith  line,  ZZ,  as 
shown  in  the  fi2:ure  at  C. 

Problem  XIIL    Conversely :  Given  a  point  by  its  per- 
To  find  tiie     spective,  a,  and  its  projection  on  the  plane  of 
of  a  gS''    the  picture,  I^,  to  find  its  co-ordinates. 
^°*°  ■  The  horizontal  and  vertical  distance  of  1^ 

from  V^,  in  the  plane  of  the  picture,  are  the  co-ordi- 
nates K  and  z,  and  the  real  length  of  the  line  l^a 
(Problem  XI.)  is  the  third  co-ordinate,  c. 


PROBLEMS   OF  PLANES,  LINES   AND    POINTS.         289 

III.  Problems  of  Planes,  Lines  and  Points. 

Problem  XIV.  Given  a  plane  by  its  horizon  and 
trace,  — 

To  draw  a  line  in  it,  either  at  random,  or  having  a 
given    direction    parallel   or  inclined   to   the  to  draw  a 

„       ,  .  , ,    ,  .  line  in  a 

plane   of   the  picture,  or  parallel  to  a  given  plane, 
line  lying  in  or  parallel  to  the  given  plane,  or  making  a 
given  angle  with  such  a  line,  and  to  find  the  vanishing- 
point  and  the  initial-point  of  the  line  so  drawn. 

In  all  these  cases  the  vanishing-point,  which  neces- 
sarily lies  in  the  horizon  of  the  given  plane  (h),  either 
may  be  assumed,  or  is  given,  or  may  be  found  by 
Problem  V. 

The  initial-point  necessarily  lies  in  the  given  initial- 
line  or  trace  (k). 

If  the  line  is  required  to  pass  through  a  given 
point,  a, — 

.*.  Draw  the  perspective  of  the  required  line  through 
the  perspective  of  the  given  point  and  the  Through  a 
vanishing-point.      The  initial-point  is  at  the  ^^^^"p^"^*- 
intersection  of  this    perspective    line  with  the   given 
initial-line  (i). 

If  the  line  is  required  to  pass  through  two  points  in 
the  plane,  or  a  finite  line  given  by  its  terminal  points,  — 

.*.  Pass  the  perspective  of  the  line  through  the  per- 
spective of  the  given  points;  its  initial-point  Through  two 
and  vanishing-point  will  lie  at  its  intersec-  ^"*"*^" 
tion  with  the  trace  and  horizon  of  the  given  plane, 
respectively,  as  at  I'  and  V. 

19 


290  MODERN   PERSPECTIVE. 

If  the  Hue  is  required  to  be  parallel  to  the  plane  of 
the    picture,  its    perspective  will   lie   parallel    to    the 
horizon  of  the  given  plane  (1),  and  its  initial-point  and 
vanishing-point  will  be  at  an  infinite  distance. 
Parallel  to  PROBLEM  XV.    ConvGrscly  '.  Givcu  a  line  by 

the  picture.    ^^^  points,  or  by  its  initial-point  and  vanish- 
ing-point, — 

To  pass  a  plane  through  it  at  random,  or  parallel  to  a 
To  pass  a  givcu  line,  or  normal  to  a  given  plane,  or  to 
through  a      the  plane  of  the  picture. 

If  the  line  is  given  by  two  points,  they  are 
given,  virtually,  as  lying  in  parallel  normal  lines 
(Prob.  XII.).  The  line,  then,  lies  in  a  normal  plane, 
and  its  initial-point  and  vanishing-point  are  easily 
found  (Prob.  XIV.)  (h,  i). 

If  the  required  plane  is  to  be  normal  to  a  plane  it 
will  be  parallel  to  the  axis  of  the  plane ;  if  normal  to 
the  plane  of  the  picture  it  will  be  parallel  to  the  axis 
of  the  picture.  These  conditions  are  comprised,  then, 
in  the  condition  that  the  required  plane  shall  contain  a 
line  given  by  its  initial-point  and  its  vanishing-point, 
Parallel  to  a  ^^^^  shall  bc  parallel  to  a  second  line,  the 
giTenime.  yanishing-poiut  of  which  is  either  given  or 
easily  ascertained  (Prob.  IV.).     Hence  :  — 

.*.  Draw  a  line  through  V,  the  vanishing-point  of  the 
given  line,  and  V,  the  vanishing-point  of  the  line  to 
which  the  required  plane  is  to  be  parallel,  for  the  hori- 
zon of  the  required  plane,  and  draw  a  second  line,  parallel 
to  this  horizon,  through  the  initial-point  of  the  giveii 
line,  for  the  required  initial-line  or  trace. 


PROBLEMS    OF   PLANES,  LINES,  AND   POINTS.         291 

If  it  is  normal  to  the  plane  of  the  picture  formal  to 
the  horizon  will  pass  through  the  centre,  Y^.      *^«pi«^"^«- 

If  the  required  plane  is  to  be  drawn  through  the 
given  line  at  random,  any  two  parallel  lines  drawn 
through  its  initial-point  and  vanishing-point 

.  At  random. 

will    represent,   respectively,   the    trace    and 

horizon  of  a  plane  passing  through  it,  as  in  Problems 

X.  and  XL  (413). 

Problem  XVI.   To  pass  a  line  through  a  point  given 
by  its  perspective,  and  lying  in  a  plane  given 
by  its   horizon   and   trace,  that  shall  lie   in  JJat^^aS- 
the  given  plane  and   shall  be   parallel  to  a  ondpi^e?" 
second  plane. 

The  required  line  will  lie  in  one  plane  and  be  parallel 
to  the  other.  Accordingly  its  vanishing-point  will  lie 
in  both  their  horizons  (h) ;  that  is  to  say,  at  the  inter- 
section of  their  horizons.     Hence :  — 

.*.  Draw  a  line  from  the  intersection  of  the  horizons 
of  the  two  planes,  as  a  vanishing-point,  through  the 
perspective  of  the  given  point  to  the  initial-line  of  the 
plane  in  w^hich  it  lies.  The  point  in  which  it  meets 
this  line  will  be  the  initial-point  of  the  required  line. 

Problem  XVL  a.  If  the  second  plane  is  parallel  to 
the  first,  the  line  can  of  course  be  drawn  in  any 
direction. 

Problem  XVL  b.  If  the  second  plane  is  parallel  to 
the  plane  of  the  picture  the  required  line  will  also  be 
parallel  to  the  plane  of  the  picture,  and  will  be  drawn 
parallel  to  the  horizon  of  the  plane  in  which  it  lies  (1). 


292  MODERN   PERSPECTIVE. 

Problem  XVI.  c.  If  the  two  planes,  though  not 
parallel,  have  those  elements  parallel  which  are  parallel 
to  the  plane  of  the  picture,  so  that  the  horizons  of  the 
two  planes  are  parallel,  as  in  Problem  VII.  B,  and 
in  Fig.  123,  Plate  XXV.,  the  required  line  will  still 
be  drawn  parallel  to  the  horizon  of  the  plane  in 
which  it  lies.  It  will  then  still  be  directed  towards 
the  infinitely  distant  intersection  of  the  two  parallel 
horizons  (1). 

Problem  XVIL   Given  a  point  by  its  co-ordinates, 
To  pass  a       ^^'  ^7  ^^^  projcctiou,  or  by  its  perspective  and 
apointT^''   a  line  or  a  plane  in  which  it  lies,— 
^^^^^'  To  pass  a  line  through  it  at  random,  or 

having  a  given  direction  parallel  or  inclined  to  the 
plane  of  the  picture,  or  parallel  to  a  given  line,  or 
normal  to  or  making  a  given  angle  with  a  given  line  in 
a  given  plane,  or  normal  to  a  given  plane,  or  to  the 
plane  of  the  picture. 

All  of  these  different  ways  of  determining  a  point  in 
position  have  been  reduced  by  the  previous  problems 
to  the  case  of  a  point  given  by  its  perspective  and  by 
the  initial-point  and  vanishing-point  of  a  line  passing 
throuQ^h  it.  So.  also,  all  the  conditions  attached  to  the 
required  line  have  been  reduced  to  the  conditions  that 
its  vanishing-point  is  known  ;  that  is  to  say,  it  is  either 
assumed,  is  given,  or  may  be  found. 

The  problem  is  then  simply  this  :  — 

Given  a  point  by  its  perspective,  a,  and  the  initial- 
point,  I,  and  vanishing-point,  V,  of  a  line  passing 
through  it, — 


PROBLEMS   OF   PLANES,  LINES,  AND   POINTS.         293 

To  pass  a  line  through  it  in  the  direction  given  by  a 
second  vanishing-point,  V,  and  to  find  I',  the  in  a  given 
initial-point  of  the  required  line.  direction. 

.'.  Pass  a  plane  through  the  given  line  parallel  to  the 
required  line  (Prob.  XV.).  The  given  point  will  lie  in 
this  plane. 

In  this  plane  draw  a  line  through  the  given  point 
having  the  required  direction  (Prob.  XIV.). 

Problem  XVIII.  To  pass  a  line  through  two  points 
given  by  their  co-ordinates,  or  otherwise,  and  to  to  pass  a 

_  line  through 

find  its  initial-point  and  its  vanishing-point.       t«'o  points. 

Each  point  lies,  or  may  be  found  to  lie,  in  a  given 
line  normal  to  the  plane  of  the  picture.  The  two 
given  points  lie,  then,  in  a  given  normal  plane,  and 
the  initial-point  and  vanishing-point  of  a  line  joining 
them  will  lie  in  the  trace  and  horizon  of  that  plane 
(Prob.  XIV.). 

Problem  XIX.   Given  three  points  by  their  co-ordi- 
nates, or  otherwise,  —  to  pass  a 
To  pass  a  plane  through  them :  through 

_  1  three  points. 

.'.  Pass   a   Ime   through   the  first  and  sec- 
ond, and  another  through  the  second  and  third  (Prob. 
XVIII.). 

Pass  the  required  plane  through  the  first  line  parallel 
to  the  second  (Prob.  XV.). 

Problem  XX.   Given  two  lines,  in  the  same  plane, 
to  pass  a  plane  through  them,  and  find  their  ^o  pass  a 
point  of  intersection.  fhrSSgh  two 

.-.  Draw  a  line    through   V    and   Y',  the 


294  MODERN   PERSPECTIVE. 

vanishing-points  of  the  two  lines,  for  the  horizon  of  the 
plane,  and  through  their  initial-points,  I  and  I',  for  its 
trace.  The  intersection  of  their  perspectives  will  be 
the  perspective  of  their  intersection. 

[The  condition  that  the  lines  lie  in  the  same  plane 
necessarily  implies  that  they  intersect  one  another, 
or  will  do  so  if  prolonged,  and  that  the  line  joining 
their  initial-points  is  parallel  to  the  line  joining  their 
vanishing-points.] 

Problem  XXL  Conversely:  to  discover  whether  two 
To  determine  ^^^^^  S^^'^n  in  perspcctivc  interscct  in  space; 
iTnes  h^terr"*"  that  is  to  Say,  whether  they  lie  in  the  same 

sect.  1 

plane :  — 

.•.  If  the  line  joining  their  initial-points  is  parallel  to 
the  line  joining  their  vanishing-points,  they  do ;  other- 
wise, not. 

Problem  XX.  a.  If  the  lines  are  parallel  to  one  an- 
To  ass  a  Other,  the  line  joining  their  initial-points  will 
through  two  ^6  the  trace  of  the  required  plane,  and  its 
para  e  ines.  j^^^j^on  wiU  be  drawn  parallel  to  this  trace 
through  their  common  vanishing-point.  There  will,  of 
course,  be  no  point  of  intersection. 

B.  If  one  of  the  lines  is  parallel  to  the  plane  of  the 
To  pass  a  picturc  it  will  be  parallel  to  the  horizon  and  to 
through  two   the  trace   oi   any  plane  that  passes  through 

lines  when 

oneisparai-    it  (1),     The  rcQuired  horizon  and  trace  will  ac- 

lel  to  the  ^   ^  i 

picture.  cordingly  be  drawn  parallel  to  this  line  through 
the  initial-point  and  vanishing-point  of  the  other  line. 

c.  If  both  lines  are  parallel  to  the  plane  of  the 
picture,  an  auxiliary  line  must  be  drawn  from  a  point 


PKOBLEMS    OF   PLANKS,  LINES,  AND    POINTS.         295 

in  the  first  line  to  a  point  in  the  second,  and  The  same 
its    initial-point    and   vanishinsj-point   found  iinesarepar- 

"^  allel  to  the 

(Prob.  XVIII.).     The  trace  and  horizon  of  the  picture, 
required  plane  may  then  be  drawn  through  the  initial- 
point  and  vanishing-point  of  this  auxiliary  line,  parallel 
to  the  given  lines. 

Problem  XXL  a.  Conversely :  to  discover  whether 
two  lines  given  in  perspective  intersect  with  to determine 
one  another,  one  being  an  oblique  line,  given  Les  hiter^^ 
by  its  initial-point,  I,  and  its  vanishing-point,  one  is  parai- 

''  ^  ^  -^  lel  to  the 

V,  and  the  other  a  line  parallel  to  the  plane  picture. 
of  the  picture,  given  in  position  by  the  projection  upon 
the  plane  of  the  picture  at  I^,  of  some  point  in  it,  a, 
the  centre,  Y^,  being  also  given. 

.*.  Draw  through  a,  the  perspective  of  the  given  point, 
its  line  of  projection  from  V^  to  I^,  and  through  the 
same  point  an  auxiliary  line,  parallel  to  the  given 
oblique  line.  These  two  lines  will  lie  in  a  normal  plane 
whose  horizon  will  pass  through  V  and  V^.  Its  trace, 
drawn  through  I*^  parallel  to  this  horizon,  will  determine 
I',  the  initial-point  of  the  auxiliary  line.  If  a  second 
plane  is  now  passed  through  the  auxiliary  line  and  the 
given  line  which  is  parallel  to  the  picture,  its  trace  T 
may  be  drawn  through  I'  parallel  to  the  given  parallel 
line.  The  auxiliary  line  and  the  parallel  line  will  both 
lie  in  this  second  plane. 

Now  if  the  given  oblique  line  really  intersects  the  given 
parallel  line  at  h,  it  also  will  lie  in  this  plane,  and  the  trace 
T  will  pass  through  its  initial-point,  I,  as  shown  at  A. 

If,  then,  the  criven  lines  intersect  one  another,  the 


296  MODERN   PERSPECTIVE. 

two  initial-points  I  and  I'  will  be  equidistant  from  the 
given  parallel  line. 

But  if  the  given  lines  do  not  intersect  at  h,  then,  as  in 
XXL  B,  the  oblique  line  will  not  lie  in  the  same  plane 
with  the  parallel  line  and  the  auxiliary  line,  and  I  will 
not  lie  in  the  trace  T. 

In  tlie  figure  the  oblique  line  is  plainly  on  the  hither 
side  of  the  parallel  line. 

Problem  XXII.  Given  two  planes,  by  their  traces 
To  find  the     and  liorizous, — 

intersection  -,      ^      ■     -,'  i>    >     , 

of  two  planes.      To  find  thcir  line  of  intersection. 

.-.  Draw  a  line  from  the  intersection  of  the  two  traces, 
'as  its  initial-point,  to  the  intersection  of  the  two  hori- 
zons, as  its  vanishing-point  (n). 

Problem  XXII.  a.  If  the  two  planes,  as  in  Problem 
VII.  B,  and  in  Problem  XVI.  c,  have  those 

The  same,  '  ' 

h^rTzonrire  elements  parallel  which  are  parallel  to  the 
parallel.  pjj^i^g  of  thc  picturc,  SO  that  their  horizons 
and  traces  are  all  parallel,  like  those  shown  in  Figs.  119 
and  123,  Plate  XXV.,  their  line  of  intersection  will  be 
parallel  to  their  traces  and  horizons,  being  directed  to 
their  infinitely-distant  points  of  intersection. 

To  find  the  position  of  this  line  an  auxiliary  plane 
must  be  passed  across  the  two  given  planes,  and  its 
lines  of  intersection  found.  The  point  in  which  these 
lines  of  intersection  meet  will  be  one  point  in  the  inter- 
section of  the  given  planes,  and  the  required  line  of 
intersection  may  be  drawn  through  the  point  parallel  to 
the  given  traces  and  horizons. 

Such  an   auxiliary   plane   will  have   its  horizon,  of 


PROBLEMS   OF   PLANES,  LINES,  AND    POINTS.        297 

course,  parallel  to  its  trace ;  and  as  such  a  plane  may- 
take  any  direction,  any  two  parallel  lines  may  be  held 
to  represent  its  horizon  and  trace.     Hence:  — 

To  find  the  required  line  of  intersection  of  two  planes 
whose  traces  and  horizons  are  all  parallel,  — 

.'.  Draw  any  line  across  the  given  traces  as  the  trace 
of  the  auxiliary  plane,  and  any  other  line  parallel  to 
the  first  across  the  given  horizons  as  its  horizon.  Draw 
tlie  two  lines  in  which  the  auxiliary  plane  intersects  the 
given  planes  (Prob.  XXIL),  and  through  the  point  where 
they  cross  one  another  draw  the  required  line  parallel 
to  the  given  traces  and  horizons. 

Problem  XXIII.    Given  three  planes,  — 

To  find  their  point  of  intersection. 

.'.  Find  the  line  of  intersection  of  the  first  to  end  the 

TIT  p    •  point  of  iu- 

and   second,  and   the  Ime  of  intersection  of  ter,=ection  of 

three  planes. 

the  second   and  third   (Prob.  XXIL).      Then 

find   the   point  of  intersection  of  these   two   lines    of 

intersection  (Prob.  XX.). 

Problem  XXIV.   Given  a  line  and  a  plane,  — 
To  find  the  point  in  which  the  line  pierces  the  plane. 
.*.  Pass   through   the    given   line,   an   aux-   Tofind 
iliary  plane,  at  random  (Prob.  XV.),  and  find  ^erces'a"''' 
its  line  of  intersection  with  the  given  plane  ^^°^' 
(Prob.  XXIL).     This  line  of  intersection,  and  the  given 
line,  will  both  lie  in  the  auxiliary  plane.     Find  their 
point  of  meeting  (Prob.  XX.).     This  point  will  lie  in 
the  given  line,  and  also  in  the  given  plane,  and  will  be 
the  point  of  puncture  required.    The  auxiliary  plane  has 


298  MODERN   PERSPECTIVE. 

been  taken  as  vertical,  this  being,  in  the  majority  of 
cases,  the  most  convenient  position  in  which  to  assume 
such  a  plane. 

Problem  XXIV.  a.  If  the  given  line  is  parallel  to 
The  same,      the  plane  of  the  picture,  its  position  is  fixed  bv 

when  the  ^  l  '  r  J 

line  is  parai-  that  of  somc  poiut  iu  it,  and  a  normal  plane 

lel  to  the  ^  '  ^ 

picture.         ^-^^y  ^q  passcd  through  it,  as  in  Problem  XV. 

IV.   Frohlems  of  Projection,  etc. 

Problem  XXV.  Given  a  point  by  its  perspective,  a, 
To  find  the  ^^^  ^^^  vanishing-point  and  initial-point  of  the 
TpSupon  normal  or  oblique  line  in  which  it  lies,  — 

To  find  its  orthographic  projection  upon  a 
given  plane. 

.'.  Find  the  vanishing-point  of  lines  normal  to  the 
given  plane  (Prob.  IV.),  and  draw  through  the  given 
point  such  a  line  (Prob.  XVIL). 

Find  the  point  in  which  this  normal  line  pierces  the 
given  plane  (Prob.  XXIV.) . 

This  point  is  the  required  projection.  The  normal 
line  is  the  line  of  projection. 

Problem  XXV.  a.  If  the  given  plane  is  parallel  to 
The  same,      the  pcrspectivc  plane  it  will  be  given  in  posi- 

when  the  i'  ^  l_^  '  o  ... 

plane  is  par-    tion  by  the  pcrspcctive  of  some  point  in  it 

alleltothe  ... 

picture.  with  its  projcction,  I^,  upon  the  plane  of  the 
picture.  The  centre,  V^,  will  be  the  vanishing-point  of 
the  line  of  projection  (q). 

.*.  Draw  the  line  of  projection  of  the  point  in  the 
given  plane,  and  also  the  line  of  projection  of  the  point 
in  the  given  line,  and  find  their  initial-points  (Prob. 


PROBLEMS    OF   PROJECTION,  ETC.  299 

XVII.).  The  two  lines  of  projection  will  lie  in  a 
normal  plane,  the  trace  of  which  will  pass  through 
their  two  initial-points.  This  normal  plane  will  inter- 
sect the  given  plane  in  a  line  parallel  to  this  trace,  and 
the  point  in  which  this  line  cuts  the  line  of  projection 
of  the  point  in  the  given  line,  is  the  point  required. 

Problem  XXVI.  Given  a  line  in  perspective,  to  find 
its  projection  upon  a  given  plane.  ^^  ^^^^^^^ 

.•.  Find  the  projection  upon  the  given  plane  rui^upon^ 
of  some  point  of  the  line  (XXV.).  ^  ^  '"^^' 

Pass  through  the  given  line  an  auxiliary  plane  normal 
to  the  given  plane  (Prob.  XV.).  Find  the  line  in  which 
this  auxiliary  plane  intersects  the  given  plane  (Prob. 
XX.).     This  line  is  the  required  line  of  projection. 

Problem  XXVII.  Given  a  point  and  a  line  in  per- 
spective, — 

To  find  upon  the  given  line  its  point  of  to  find  the 

distance  be- 

nearest  approach  to  the  given  point,  and  the  tweenapoint 
distance  between  them. 

.*.  Pass  through  the  given  point  a  plane  normal  to 
the  given  line  (Prob.  III.),  and  find  the  point  in  which 
the  given  line  pierces  this  plane  (Prob.  XXIV.).  This 
point  is  obviously  the  required  point,  and  its  dis- 
tance from  the  given  point  (Prob.  XI.),  the  required 
distance. 

Problem  XXVIII.   Given  two  lines  in  perspective, 
to  find  their  points  of  nearest  approach,  and  to  find  the 
the  distance  and  direction  from  one  another  ?weea?w?' 
of  those  points. 


300  MODERN   PERSPECTIVE. 

.*.  Pass  through  one  line,  I'  Y',  an  auxiliary  plane 
parallel  to  the  other,  I V,  (Prob.  XV.),  and  find  the  pro- 
jection of  the  second  line  upon  this  plane,  to  which  it 
is  parallel  (Prob.  XXVI.).  This  projection  and  the  first 
line  will  both  lie  in  this  plane. 

Find  the  point  a  in  which  the  projection  of  the  second 
line  intersects  the  first  line  (Prob.  XX.).  This  is  one  of 
the  required  points,  and  the  point  6  of  the  second  line 
thus  projected  is  the  other. 

Find  the  length  of  the  line  a  h  joining  these  two  points 
(Prob.  XL) ;  this  is  the  required  distance  of  the  two 
points,  and  the  direction  of  the  line  joining  them,  which 
is  the  direction  of  lines  normal  to  the  auxiliary  plane,  is 
the  required  direction. 

Problem  XXIX.   Given  the  perspective  of  a  finite 
line,  and  its  direction  by  the  vanishing-point  of  the  line 
in  which  it  lies  :  — 
To  divide  a         To  dividc  it  in  any  desired  ratio,  or  into 

line  in  any  •  n 

given  ratio,     equal  parts,  or  proportionally  to  another  line. 

[This  problem  is  analogous  to  the  problem  of  cut- 
ting a  giA^en  length  from  a  perspective  line  (Prob.  X.). 
But  in  that  case  the  given  line  is  indefinite  in  length, 
and  the  auxiliary  triangle  is  an  isosceles  triangle,  of 
which  the  adjacent  sides  are  divided  into  equal  parts  by 
lines  drawn  parallel  to  the  base.  In  the  present  case 
the  perspective  line  is  of  definite  length,  and  the  aux- 
iliary triangle  is  scalene,  the  adjacent  sides  being  divided 
proportionally.  ] 

.*.  Through  the  given  vanishing-point,  draw  a  line  in 


PROBLEMS    OF   PllOJECTlON,  ETC.  301 

any  coDvenient  direction  as  the  horizon  of  a  plane 
passing  through  the  given  line  (j). 

Through  either  end  of  tlie  given  finite  line  draw  a  line 
parallel  to  the  horizon  thus  assumed,  and  lay  off  upon 
this  line,  as  a  line  of  measures,  the  given  proportional  or 
equal  parts  at  any  convenient  scale. 

Draw  through  the  extreme  point  thus  ascertained  a 
line  passing  through  the  other  end  of  the  given  finite 
line,  and  prolong  it  until  it  intersects  the  assumed  hori- 
zon. Towards  this  point  of  intersection,  as  a  vanishing- 
point,  draw  lines  from  the  points  taken  upon  the  line  of 
measures.  The  points  at  which  they  cut  the  given  line 
will  be  the  required  points  of  division. 

Problem  XXIX.  a.  If  the  given  line  is  parallel  to 
the  plane  of  the  picture,  its  perspective  will  The  same, 

.,.,,.  ,  .  when  the 

be  proportional  m  all  its  parts  to  the  given  uneispar- 

^       ^  ^  °  alleltothe 

line,  and  it  may  be  divided,  as  the  line  itself  picture, 
would  be,  according  to  the  methods  of  plane  geometry. 


SUMMAEY. 

These  twenty-nine  Geometrical  Problems  may  more 
briefly  be  stated  as  follows  :  — 

Besides  the  data  here  mentioned,  the  position  of  the 
centre,  V^,  and  the  length  of  the  Axis,  C^,  which  fixes 
the  position  of  the  station-point,  S,  are  understood  to 
be  given,  though  these  are  not  needed  for  the  solution 
of  the  problems  numbered  from  XII.  to  XXIV.,  nor 
for  Problem  XXIX. 


302 


MODERN  PERSPECTIVE. 


FroUems  of  Direction, 


Problem. 

I. 

11. 

III. 

IV. 

V. 

V.  A. 

VI. 

VI.  A. 

VII. 

VII.  A. 
VII.  B. 

VIII. 
IX. 


Given. 

a,/3. 

a,y. 

V. 

H. 

H,  V,( 


Requibed. 
V. 
V. 
H. 

VT 

v. 


The  same,  when  V  is  at  infinity. 

V,  v.  <t>. 

The  same,  when  V  is  at  infinity. 

H,  H'.  <t>. 

The  same,  when  the  planes  are  normal. 

The  same,  when  H  and  H'  are  parallel. 

H,  V.  <l>. 

v.  The  length  of  the  optical  line,  and  locm  of  D. 


Problem. 

X. 

X.  A. 

XI. 
XL  A. 

XII. 
XIII. 


Problems  of  Dimension  and  Position. 

Given.  Required. 

IM,  VM ;  m.  M. 

The  same,  when  the  given  line  is  parallel  to  pp,  I  and  V  being  at  an 

infinite  distance. 
IM,  VM  ;  M.  m. 

The  same,  when  m  is  parallel  to  pp,  I  and  V  being  at  an  infinite 

distance. 
K,  c,  iz  ;  VC.  a. 

a,ic,  yc.  K,  c,  z. 


ProUems  of  Lines,  Planes,  and  Points. 


Problem. 

XIV. 

XIV.  A. 

XV. 

XV.  A. 

XVI. 

XVI.  A. 
XVI.   B. 

XVII. 

XVIII. 

XIX. 

XX. 


Given. 
H,  T ;  a. 
H,  T  ;  a,  a'. 
I,  V ;  or  a,  a'. 


Required. 
V,I. 
V,I. 
H.T. 


The  same,  when  the  required  plane  is  to  be  parallel  or  normal  to  the 

given  line,  or  parallel  to  pp. 
H,  T  ;  a,  H'.  I,  V. 

The  same,  when  H'  is  parallel  to  H. 
The  same,  when  I,  V,  and  H'  are  at  an  infinite  distance, 
c,  I ;  V,  v.  I'. 

a,  a'.  I,  V. 

a,  a',  a".  T,  H. 

T,  H :  I,  V  ;  I',  V.    The  point  of  intersection  of  the  two  lines. 


PROBLEMS    OF   PROJECTION,   ETC. 


303 


Problem. 
XX.  A. 


XXI. 

XXI.  A. 

XXII. 

xxn  A. 

XXIII. 

XXIV. 

XXIV.  A. 


Given.  Required. 

The  same,  when  V=V' ;  when  V  is  at  an  infinite  distance  ;  when  V 

and  V  are  both  at  an  infinite  distance  ;  i.  e.,  when  we  have  one 

line  or  both  parallel  to  pp. 
I,  V  ;  I',  v.  Do  the  lines  intersect  ? 

The  same,  when  we  have  one  line  or  both  parallel  to  pp. 
T,  H ;  T',  H'.  The  line  of  intersection  of  the  two  planes. 

The  same,  when  H  and  H'  are  parallel. 
T,H;  T',H';  T",H".  The  pomt  of  intersection. 
T,  H  ;  I,  V.  The  point  of  intersection. 

The  same,  when  I  and  V  are  at  an  infinite  distance ;  i.  e.,  when  the 

line  is  parallel  to^^. 


ProUems  of  Projection. 


Problem.  Given.  Required. 

XXV.  I,  V  ;  a;  T,  H.  The  projection  of  the  point  on  the  plane. 

XXV.  A.  The  same,  when  T  and  H  are  at  an  infinite  distance  ;  i.  e.,  when  the 
plane  is  parallel  io pp. 

XXVI.  I,  V  ;  T,  H.  The  projection  of  the  line  on  the  plane. 

XXVII.  I,  V  ;  a.  The  distance  of  the  point  from  the  line. 

XXVIII.  I,  V  ;  I',  v.  The  distance  apart  of  the  two  lines. 

XXIX.  VM,  M.  To  divide  m  in  a  given  ratio. 

XXIX.  A.  The  same,  when  the  line  is  parallel  to  pp. 


CHAPTER  XIX. 

THE   PRACTICAL  PROBLEM. 

433.   After  all,  the  question  remains,  How  is  one  to 
go  to  work,  in  a  given  case,  to  make  a  per- 


The  data. 


spective  drawing.  The  shape  and  size  of  the 
object  to  be  drawn,  a  building  for  instance,  are,  of 
course,  supposed  to  be  given,  with  the  scale  to  be  em- 
ployed in  the  plane  of  the  picture.  The  scale  either 
may  be  assumed  or  may  be  determined  by  comparing 
the  relative  distances  of  the  object  and  of  the  plane  of 
the  picture  from  the  spectator  (94). 

434.  The  next  thing  to  be  determined  is  the  attitude 
The  attitude  ^^  ^^®  objcct ;  that  is  to  say,  the  angle  its 
of  the  object.  p^,ij3(3ipai  li^es  shall  make  with  a  line  drawn 
from  the  eye  to  the  object.  The  direction  of  this  line 
is  in  general  purely  arbitrary,  being  so  chosen  as  to 
exhibit  the  building  or  other  object  in  its  best  aspect. 
The  plane  of  the  picture  is  generally  taken  at  right 
angles  to  this  line,  which  then  becomes  the  Axis  of 
the  picture,  some  point  near  the  middle  of  the  object 
being  then  at  the  Centre.  But  when  it  is  possible,  by 
giving  this  Axis  a  slightly  different  direction,  to  bring 


THE  PRACTICAL  PROBLEM.  305 

the  principal  lines  of  the  object  at  45°  with  the  plane 
of  the  picture,  making  the  Centre,  V^,  coincide  Best  at  45' 
with  V^,  the  "vanishing-point  of  45°,"  it  is 
best  to  do  so.    This  adjustment  is  exemplified  in  Fig.  14, 
Plate  v.,  and  in  Fig.  138,  Plate  XXVII. 

435.  When  one  side  of  the  object  is  nearly  parallel 
with  the  picture,  it   is   often   desirable,  and 
sometimes  necessary,  as  has  been  explained  ^'"p"^"^^- 
in  paragraph  276,  to  make  it  exactly  parallel. 

436.  It  is  generally  desirable  to  have  the  plane  of  the 
picture  as  far  from  the  eye  as  possible,  which  The  Axis  to 

be  as  long 

is  equivalent,  for  any  given  scale,  to  having  as  possible, 
the  object  itself  as  distant  as  possible,  the  distance 
of  the  picture  multiplied  by  the  denominator  of  the 
fraction  expressing  the  scale,  giving  the  distance  of  the 
object  (94).  If,  for  instance,  the  scale  employed  in 
the  drawing  is  that  of  one  eighth  of  an  inch  to  the  foot, 
that  is  to  say,  one  ninety-sixth  full  size,  the  object,  or 
that  portion  of  it  that  lies  in  tli^e  plane  of  projection,  or 
plane  of  measures,  will  be  ninety-six  times  as  far  away 
as  the  drawing. 

The  further  the  station-point  is  from  the  picture  the 
easier  will  it  be  for  the  spectator  to  occupy  it,  and  the 
less  will  be  the  apparent  distortion  of  the  drawing  if  he 
fails  to  occupy  it  exactly. 

437.  But  setting  the  station-point  far  from  the  pict- 
ure is  equivalent  to  setting  the  vanishing-  ^  ^  vLand 
points  far  apart,  so  that  practically  the  first  Irlpart 

thing  to  do  after  the  attitude  of  the  object  is 

20 


VR  to  be  as 
far  apart 
possible. 


306  MODERN   PERSPECTIVE. 

chosen,  and  the  angle  it  is  to  make  with  the  plane  of  the 
picture  determined,  is  to  fix  the  distance  apart  of  the 
principal  right-hand  and  left-hand  vanishing-points,  Y^ 
and  V^  These  points,  which  of  course  lie  in  the  Hori- 
zon, are  generally  set  at  the  extreme  limits  of  the  table 
or  drawing-board  upon  which  the  work  is  to  be  done. 
See  Fig.  137,  A,  Plate  XXVII.,  in  which  the  assumed 
attitude  of  the  object  is  shown  by  two  lines  drawn  at 
right  angles  to  one  another,  making  the  given  angles 
witli  the  picture.  These  lines  may  be  drawn  in  any 
convenient  place,  it  makes  no  difference  where. 

438.  As  the  principal  horizontal  lines,  E  and  L,  van- 
piate  XXVII.  ishing  at  V^  and  V^,  upon  the  Horizon,  are 
Fig.  i3(.  generally  at  right  angles,  the  station-point  S 
is,  in  plan,  generally  at  the  vertex  of  a  right-angled 
triangle,  of  which  the  line  V^  \^,  in  the  plane  of  the 
picture,  is  the  hypotenuse.  The  locus  of  S  is  accordingly 
a  horizontal  semicircle,  of  which  the  line  V^  V^  is  the 
diameter.  The  next  step  after  fixing  these  points  is, 
then,  to  describe  such  a  semicircle,  and  to  find  upon  it 
s.vc,vx,      the  point  S,  such  that  the  line  SV^  will  be 

DL,  DR,  and  ^ 

D^-  parallel  to  the  right-hand  side  of  the  object, 

and  SV^  to  its  left-hand  side.  A  perpendicular  dropped 
from  this  point  upon  the  line  V^  V^  will  give  the  posi- 
tion of  the  Centre,  V^ ;  a  diagonal  line  bisecting 
the  right  angle  will  give  the  point  V^,  the  "  vanishing- 
point  of  45°  "  (44) ;  and  the  lines  SV«  and  SV^  revolved 
into  the  plane  of  the  picture  will  give  respectively 
the  right  and  left-hand  points  of  distance,  D^  and  D^. 


THE  PRACTICAL   PROBLEM.  307 

D^,  the  point  of  distance  of  the  diagonal  line,  may 
be  obtained  at  the  same  time,  if  desired,  by  revolv- 
ing SV^  into  the  plane  of  the  picture,  as  in  Plate  IV., 
Fig.  11. 

439.  If  shadows  are  to  be  cast,  and  the  vanishimr. 
point  of  shadows,  V^,  lies  beyond  either  of  the  principal 
vanishing-points,  as  it  does  in  the  figure,  room  must  be 
allowed  for  this  vanishing-point  also.  This  space,  how- 
ever, may  be  saved  by  taking  the  sun  in  the  plane  of  the 
picture,  as  in  Fig.  36,  Plate  VIII.,  with  the  vanishing- 
point  of  shadows  at  an  infinite  distance  (184). 

440.  If  the  object  is  set  just  at  45°,  as  has  been  rec- 
ommended, and  as  is  done  in  Plate  V.,  and  in  The  object  at 

45°. 

Fig.  138,  Plate  XXVIL,  its  two  sides  making  Fig.  i38. 
equal  angles  with  the  plane  of  the  picture,  it  is  not 
necessary  to  describe  the  semicircle  at  all.  The  Centre 
will  be  half  way  between  V^  and  V^,  the  station-point 
will  be  the  same  distance  in  front  of  the  Centre,  V-^  wiU 
coincide  with  V^,  and  D^  and  D^  may  be  found  as 
before.  They  will  be  ahnost  exactly  two  fifths  of  the 
distance  from  the  Centre  to  either  vanishing-point,  as 
shown. 

441.  All  these  operations  are  conducted  in  plan,  the 
paper  at  137,  A,  and  138,  A,  representing  the  The  ground 
ground  plane,  or  horizontal  plane  of  projec-  ^^*"®" 
tion,  the  line  V^  V^  being  the  projection  of  the  Horizon. 
This  line  also  represents  both  the  plane  of  the  picture, 
pp,  seen  edgewise,  and  the  ground  line,  g  I,  in  which  the 
plane  of  the  picture  cuts  the  ground  plane. 

442.  If  now  the  plane  of  the  picture  is  revolved  into 


308  MODERN  PERSPECTIVE. 

the  plane  of  the  paper  about  the  horizon,  the  points 
rr,     1       ,  V^  V^  V^  D^  D^  and  V^  will  retain  their 

The  plane  of  »  j  >  ^  ? 

the  picture.  pQgj|^ions,  the  ground  line,  g  I,  will  appear  in 
the  plane  of  the  paper  at  some  distance  below  the 
Horizon,  and  parallel  to  it,  and  tlie  station-point,  S,  will 
be  in  the  air  in  front  of  the  picture,  opposite  the  Centre, 
as  shown  in  Plate  XXV.,  and  in  Fig.  13,  Plate  IV. 

443.  Vertical  lines,  erected  in  the  plane  of  the  picture 
VM,  vM',  ^^  ^^  ^^^^  ^^'  ^'ill  ^0^^^  establish  HPtZ  and  HLZ, 
VN,  audVN'.  ^y^^  horizons  of  the  principal  vertical  planes, 
and  the  vanishing-points  of  the  inclined  lines  M,  M',  N, 
and  N',  lying  in  or  parallel  to  these  planes, may  be  fixed  by 
drawing  lines  at  D^  and  D^,  that  make,  with  the  Horizon, 
the  same  angles,  ^  and  /3',  that  the  lines  themselves  make 
with  the  horizontal  plane.  The  points  in  which  these 
lines  intersect  the  horizons  of  the  vertical  planes  will 
give  the  vanishing-points  V^,  V^',  V^,  and  V^'  (88). 
If,  as  is  generally  the  case,  the  roofs  have  the  same 
slope,  yS  and  ^'  will  be  equal. 

444.  These  points  being  determined,  the  horizons  of 
HRN,  HLM,  inclined  planes,  HRN,HRN' HLM,  and  HLM', 

etc.,  V^  and  ^  -,     .  i 

v^'.  can  be  drawn  whenever  they  are  needed,  and 

V^  and  V^',  the  "vanishing-points  of  hips  and  valleys," 
can  be  ascertained,  as  in  Plates  L,  11.,  III.,  IV.,  and  V. 

445.  In   the  same  way  V^,  the   vanishing-point   of 

sunlight  or  of  shadows  may  be  fixed  by  deter- 
mining first  a  point,  V^'  (or  V^'),  the  vanishing- 
point  of  the  horizontal  projection  of  the  rays  of  light, 
as  in  Fig.  137,  A.     D^'  (or  D^')  can  then  be  obtained 
by  revolving   the  horizontal   line   SV^'  (or  SV^)'  into 


THE  PRACTICAL  PEOBLEM.  309 

the  plane  of  the  picture.  The  position  of  V^,  imme- 
diately above  or  below  V^'  (or  Y^'),  can  then  be  fixed 
by  laying  off  at  D^'  (or  D^')  the  real  angle  made  by 
rays  of  light  with  the  ground-plane,  and  the  horizons 
of  the  planes  of  invisible  shadows,  and  the  vanishing- 
points  of  the  lines  of  visible  shadow  obtained  by  draw- 
ing lines  from  Y^  to  the  various  vanishing-points,  and 
noting  their  intersection  with  the  horizons  previously 
deterniined,  as  is  done  in  Figs.  34,  35,  36,  and  38, 
Plate  YIII.  (181). 

446.  The  station-point,  S,  and  the  principal  vanishing- 
points,  Y^  and  Y^  being  once  established,  all  preliminary 
these  other  horizons,  vanishing-points,  and  ^p^'^^'^'^'^^- 
points  of  distance  may  be  determined  either  at  once,  as  in 
Fig.  38,  before  the  drawing  is  begun,  or  from  time  to 
time,  as  they  are  needed,  during  the  progress  of  the 
work.  In  either  case  they  are  the  necessary  scaffold- 
ing, so  to  speak,  without  which  the  various  construc- 
tions cannot  be  carried  on. 

447.  If  the  drawing^  is  to  be  made  in  Parallel  Per- 
spective  the  extreme  vanishing-points  become  p^^^^gj 
the  vanishing-points  of  diagonals,  which  will  p^'^'^p^^^tive. 
coincide  with  the  points  of  distance,  as  in  Plate  YI. 
It  will  generally  he  convenient  to  establish  at  once 
points  of  half  or  quarter  distance,  as  shown  in  Fig.  21, 
in  that  Plate  (142),  and  in  Fig.  94,  Plate  XX.  (333). 

448.  All  this  preliminary  w^ork  is  concerned  solely 
with  the  direction  of  lines  and  planes,  not  with  The  position 

of  the  object, 

their  position.     Before  constructing  a  perspec-  horizontally. 


310  MODERN   PERSPECTIVE. 

tive  drawing  by.  their  aid  it  is  necessary  to  determine 
also  the  position  to  be  assigned  to  the  object;  that  is  to 
say,  to  some  prominent  point  in  it.  The  point  generally 
selected  is  the  lower  end  of  the  nearest  corner. 

The  first  thing  to  be  done  is  to  determine  how  far  to 
the  right  or  left  of  the  Centre  this  point  shall  be  set. 
It  is  generally  on  the  right  if  the  left-hand  side  of  the 
object  is  to  be  made  prominent,  and  vice  versa.  In  Fig. 
137  the  position  of  the  front  corner  is  assumed,  and,  the 
building  being  rather  a  large  one,  it  is  set  considerably 
to  the  right  of  the  Centre.  In  Tig.  138  both  the  attitude 
and  the  position  of  the  building  are  determined  upon 
the  orthographic  plan  at  A. 

449.  As  everything  in  a  perspective  drawing  is  more 
or  less  distorted,  according  as  it  is  more  or  less  removed 
from  the  Centre,  opposite  the  eye  (260),  it  is  of  the 
first  importance  that  the  object  shall  be  as  near  the 
Centre  as  possible.  The  Centre  does  not  necessarily,  as 
has  been  said,  coincide  with  the  middle  of  the  picture. 
But  it  is  desirable  to  have  it  do  so,  as  nearly  as  is  con- 
venient ;  and,  as  the  object  itself  naturally  occupies  the 
middle  of  the  picture,  if  the  Centre  is  in  the  middle  of 
the  object  this  point  also  is  generally  attained. 

450.  But  it  is  often  worth  w^hile  to  throw  the  object 

considerably  to  one  side  of  the  Centre  ;  that  is 

The  Centre.  ''  ' 

to  say,  to  have  the  Centre  quite  out  of  the 
middle  of  the  picture,  if  by  so  doing  the  practical  ad- 
vantages and  convenience  of  having  the  principal  right 
and  left-hand  vanishing-points  at  45°  can  be  secured. 


THE  PRACTICAL  PROBLEM.  311 

In  Fig.  15,  for  instance,  Plate  V.,  the  Centre  is  near  the 
left-hand  corner  of  the  little  house.  But  the  main  lines 
of  the  building  do  not  materially  differ  from  those  of 
Fig.  12,  Plate  IV.,  in  which  the  Centre  is  near  the  right- 
hand  corner. 

451.  Where,  as  in  Fig.    15,  two  objects  are  to  be 
shown,  it  is  of  course  impossible  that  both 

should  occupy  the  same  place  at  the  Centre. 
But  it  is  generally  practicable  to  have  one  of  them,  or 
botli  of  them,  if,  as  in  this  case,  they  are  parallel  to  one 
another,  at  45°  with  the  picture. 

452.  The  position  to  be  given  to  the  object,  horizon- 
tally, having  been  determined,  the  next  thing 

is  to  draw  a  Perspective  Plan  of  it;  i.e.,  to  SJe'pTan.'''"" 
put  into  perspective  its  horizontal  projection. 

453.  The  horizontal  plane  upon  wliich  the  perspective 
plan  of  the  object  to  be  represented  is  sup-  ^^^^  ^^^^^ 
posed  to  be  drawn,  is  called  the  Ground-Plane,  ^'^®'^^- 
and  its  initial  \me,i.e.,  the  trace  in  which  the  ground-plane 
cuts  the  plane  of  the  picture,  is  called  the  ground-line, 
or  line  of  horizontal  measures,  as  has  been  said.  It  is 
convenient,  for  many  reasons,  to  have  this  as  far  as  may 
be  below  the  Horizon  (46,  101),  and  it  is  well  to  draw  it 
upon  a  separate  piece  of  paper,  covering  the  lower  part 
of  that  upon  which  the  drawing  is  to  be  made.  The  sunk- 

.  ,.  ,  ,.       .        .         perspective 

SO  that  the  construction  lines  that  lie  in  its  plan. 
neighborhood  may  not  deface  the  picture,  and  so  that 
they  may  be  removed  and   used   again,  if  necessary, 
instead  of  being  erased. 


312  MODERN   PERSPECTIVE. 

454.  This  is  shown  in  Figs.  137,  B,  and  138,  B,  in' 
which  the  Horizon,  with  the  various  vanishing-points 
and  points  of  distance,  are  transferred  directly  from 
Figs.  137,  A,  and  138,  A,  and  the  ground-line,  g  I,  drawn 
in  an  inch  or  two  lower  down. 

In  practice  the  figures  A  and  B  would  be  drawn  one 
over  the  other,  on  the  same  paper.  But  in  the  plate  the 
constructions  by  which  the  position  of  the  vanishing- 
points  and  points  of  distance  is  determined,  both  upon 
the  horizontal  plan  and  in  the  plane  of  the  picture,  are 
drawn  at  A,  and  the  perspective  itself,  and  the  perspec- 
tive of  the  plan,  both  of  which  are  in  the  plane  of  the 
picture,  are  drawn  at  B.  This  avoids  confusion,  and 
allows  all  the  construction  lines  to  be  shown  in  full, 
which  in  practice  would  be  drawn  only  in  part. 

455.  It  is  customary  to  have  the  front  corner  of  the 
building,  or  otlier  object  to  be  drawn,  lie  in  the  plane 
of  projection,  or,  which  comes  to  the  same  thing,  to 
The  starting-  ^^^^®  ^^^^  imaginary  model  touch  the  plane  of 
P"^'"'-  the  picture  (396)  as  in  these  figures.  In  the 
perspective  plan,  then,  the  horizontal  projection  of  this 
corner  will  lie  in  the  ground-line,  as  shown  at  the 
point  I.  Lines  drawn  from  this  point,  as  an  initial- 
point,  to  the  principal  horizontal  vanishing-points,  V^ 
and  V^  are  the  front  lines  of  a  perspective  plan.  They 
are  infinite  lines,  upon  which  the  horizontal  dimensions 
of  the  object  can  be  cut  off  by  means  of  the  points  of 
distance  already  established,  the  ground-line  serving  as 
a  line  of  horizontal  measures. 


THE  PRACTICAL  PROBLEM.  313 

456.  The  lenoth  of  the  riojht-hand  side  of  the  build- 
ing,  or  other  object,  with  its  subdivisions,  being  Horizontal 
then  laid  off  upon  the  ground-line  to  the  right  L^Sl^ptr** 
of  this  point,  and  of  the  left-hand  side  towards 

the  left,  may  be  transferred  to  these  infinite  perspective 
lines  by  drawing  lines  across  them  to  the  right-hand  and 
left-hand  points  of  distance  respectively. 

457.  In  drawing  the  perspective  plan  constant  use 
may  be  made  of  the  diagonal  line  bisecting         ^ 
the   right-angle,    and   of   its   vanisliing-point, 

Y^,  for  finding  the  plan  of  hips  and  valleys  and  other 
lines  lying  at  45°  with  the  principal  lines.  Further 
illustrations  of  this  may  be  found  in  Plates  II.  and 
III. 

If  the  principal  horizontal  lines  of  the  perspective  plan 
lie  at  45 '^  with  the  ground-line,  as  in  Fig.  138,  one  set  of 
the  right-angles  in  which  they  meet  will  be  bisected 
by  lines  drawn  to  the  Centre,  and  the  others  by  lines 
drawn  parallel  to  the  Horizon.  The  hips,  also,  on  the 
right  and  left  of  the  roofs,  will  be  parallel  to  the 
picture,  and  will  be  drawn  parallel  to  the  horizons 
of  the  planes  in  which  they  lie,  as  shown  in  the 
figure. 

458.  Dimensions  taken  by  scale  upon  the  ground- 
line  may  be  transferred  to  lines  lying  in  the  j^^^^^  ^^^^.^^ 
horizontal  plane  and  parallel  to  the  plane  of  ^^^*°^^- 
the  picture,  and  accordingly  parallel  to  the  ground-line, 
by  drawing  lines  to  any  point  on  the  Horizon  as  a  van- 
ishing-point of  parallel  lines.  The  fence  in  Fig.  137,  B, 
is  drawn  in  this  way. 


314  MODERN   PERSPECTIVE. 

459.  As  many  different  perspective  plans  may  be 
Several  per-    made  as  the  complexity  of  the  subject  may 

spective 

plans.  seem  to  require,  and   they  may  be  above  or 

below  the  picture,  as  may  be  most  convenient,  as  in 
Plates  III.  and  IV.  This  is  illustrated  also  in  Plate 
XXVIL,  where  Fig.  140  shows  three  perspective  plans, 
and  Fig.  137  two.  The  work  upon  them  may  be  done 
all  at  once,  or  from  time  to  time  during  the  progress 
of  the  drawing,  as  may  be  preferred.  All  the  details 
of  tlie  plans  of  every  part  may  thus  be  put  into  per- 
spective. But  it  is  not  of  course  necessary  to  com- 
plete the  plan  of  any  parts  that  cannot  be  seen.  In 
general  it  suffices  to  make  the  plan  of  the  two  sides 
that  show,  and  of  such  more  remote  portions  as  are 
visible  above  these  sides. 

460.  The  perspective  plan  being  made,  or  at  any  rate 
Theperspec-  ^^"^^7  bcguu,  the  drawing  itself  may  be  com- 
*^^®'  menced.  The  perspective  of  the  object  itself 
lies  directly  above  the  plan,  but  how  far  above  depends 
upon  the  relative  altitude  of  the  object  and  of  the 
The  position  spcctator.     The  points  on  a  level  with  the  eye 

of  the  object,         .,,        ,  ^  ■■  ,,         ^^ 

vertically.  wiU  always,  ot  course,  be  seen  on  the  Ho- 
rizon. So  much  of  the  object  as  is  above  the  eye,  per- 
haps the  whole  of  it,  will  appear  above  the  Horizon; 
whatever  is  below  the  eye  will  be  drawn  below  the 
The  starting-  Horizou.  The  starting-point,  that  is  to  say, 
P""'"*-  the  lower  end  of  the  front  corner,  will  lie 

dirertly  above  the  corresponding  point  in  the  perspec- 
tive plan,  and  as  far  below  the  Horizon,  by  scale,  as  the 


THE  PRACTICAL  PROBLEM.  315 

spectator's  eye  is  supposed  to  be  above  the  point  itself, 
as  at  c,  Fig.  137.  The  same  perspective  plan  will  serve 
to  make  several  views  of  the  same  object,  taken  at 
different  levels,  above  or  below  the  starting-point,  as  is 
done  in  Figs.  4,  5,  and  6,  Plate  II. 

461.  The  perspective  plan,  drawn  in  the  plane  of  the 
picture,  sufi&ces  to  determine  all  horizontal  dimensions ; 
that  is  to  say,  the  position  of  all  vertical  lines. 

The  position  of  horizontal  lines  is  determined  by 
laying  them  off  upon  the  initial-line  or  trace  Theiineof 
of  some  vertical  plane,  as  a  line  of  vertical  measures, 
measures.  When  the  nearest  corner  touches  the  plane 
of  the  picture  it  is  generally  used  for  this  purpose. 
This  line  is  then  at  once  the  initial-line  of  the  ris^ht- 
hand  vertical  plane,  E  Z,  and  of  the  left-hand  vertical 
plane,  L  Z,  and  serves  as  a  line  of  vertical  measures  for 
both,  as  at  VI?  in  Figs.  137,  B,  and  138,  B.  The  scale 
employed  is  the  same  as  that  used  upon  the  ground- 
line  for  determining  the  horizontal  dimensions  of  the 
perspective  plan,  since  all  lines  in  the  plane  of  projec- 
tion are  drawn  to  the  same  scale  (94). 

462.    But  any  plane  occurring  in  the  object  may  be 
prolonged  until  it  cuts  the  perspective  plane,  g^^^^i  ^^^^ 
and  have  a  line  of  measures  of  its  own,  in  its  '*°®^ 
own  initial-line,  as  at  v'  v'  in  Fig.  137,  B,  which  serves 
as  an  independent  line  of  measures  for  the  end  of  the 
wing.      The    vertical  dimensions  taken  upon  ygrticai 
these  lines  of  measures  may  be  transferred  di-  <*™«°^'«°^ 
rectly  to  any  vertical  line  which  lies   in  this  vertical 
plane,  and  which  is  acco^-dingly  parallel  to  the  line  of 


316  JVILODERN   PERSPECTIVE. 

vertical  measures,  by  means  of  the  vanishing-points  V^ 
and  V^.  In  this  way  is  determined  the  position  of  all 
the  horizontal  lines  in  Figs.  137,  B,  and  138,  B,  the  ver- 
tical lines  erected  from  the  corresponding  lines  in  the 
perspective  plans  serving  to  determine  their  length. 

463.  Moreover,  just  as  such  dimensions  were  trans- 
ferred to  lines  lying  in  the  ground-plane,  but  inclined 
to  the  picture,  by  means  of  a  point  of  distance  upon  the 
Horizon,  so  dimensions  taken  by  scale  upon  a  vertical 
line  of  measures  may  be  transferred  to  lines  that  lie  in 
Points  of  dis-  ^^^®  vertical  plane  and  are  inclined  to  the  plane 
vertfcai^*^'^     of  tlic  plcturc,  by  uicaus  of  points  of  distance 

taken  upon  the  horizon  of  the  vertical  plane  in 
which  they  lie  (113).  This  is  illustrated  in  Fig.  12, 
Plate  IV.,  where  the  heights  of  the  gable  are  set  off 
upon  the  vertical  line  through  the  corner  of  the  house. 

464.  If  any  part  of  the  object  advances  in  front  of  the 
Construe-  pHHcipal  Vertical  planes,  or,  in  plan,  in  front 
front  o"f  the  of  the  principal  lines  of  the  perspective  plan, 
plane.  as  Is  thc  casc  with  the  wing  of  the  building 

Fig  1.39 

shown  in  Fig.  137,  its  plan  can  be  drawn  in 
perspective  by  prolonging  tlie  leading  perspective  lines 
in  front  of  the  perspective  plane,  as  is  shown  in  Fig. 
139.  In  this  figure  the  dimensions  to  be  set  off  upon 
this  part  of  a  left-hand  line,  L  (or  of  a  right-hand  line,  R), 
are  set  off  upon  the  ground-line  to  the  right  of  the 
initial-point  instead  of  to  the  left  (or  to  the  left  instead 
of  to  the  right)  ;  and  in  transferring  them  to  the  per- 
spective line  they  are  brought  forward  away  from  the 


THE   PRACTICAL   PROBLEM.  317 

point  of  distance,  instead  of  being  carried  backward 
toward  it,  as  before. 

The  length  of  the  wing  of  the  building  in  Fig.  137 
is  ascertained  in  this  way :  The  dimension  R^,  taken 
from  the  elevation  above,  is  laid  off  upon  the  ground- 
line  to  the  left  of  the  point  I,  the  initial-point  of  the 
perspective  line,  R,  and  is  transferred  to  the  prolonga- 
tion of  that  line  in  front  of  the  plane  of  projection  and 
below  the  ground-line  by  means  of  the  point  of  distance, 
D^  as  in  Fig.  139.     See  also  Fig.  15,  Plate  V.  (109). 

Another  way  of  drawing  such  objects,  or  parts  of  an 
object,  is  shown  in  the  second  perspective  plan  at  the 
bottom  of  the  same  figure.  The  point  a,  w^here  the  wing 
joins  the  main  building,  having  been  ascertained  as  be- 
fore, by  measuring  off  upon  the  principal  left-Band  line 
the  distance,  L\  a  right-hand  line,  directed  towards  the 
riglit-hand  vanishing-point,  V^,  is  drawn  through  the 
point  a  until  it  intersects  the  ground-line  at  h,  its  initial- 
point.  If  now  a  second  line  be  drawn  through  a,  directed 
upon  D^,  the  right-hand  point  of  distance,  and  cutting  the 
ground-line  at  d,  the  distance,  hd,  intercepted  upon  the 
ground-line,  wall  be  the  real  length  of  the  line  a  h,  and 
the  real  length  of  the  wing,  R^,  may  be  laid  off  upon  the 
ground-line  from  d  and  transferred  to  the  line  ah  hy 
means  of  D^,  as  shown. 

If  parts  of  the  object  to  be  drawn  are  advanced  not 
only  in  front  of  the  principal  planes,  but  in  front  of  the 
plane  of  projection,  as  often  happens  with  the  cornices 
of  buildings,  and  with  steps  and  platforms,  as  is  shown 
in  Fig.  138,  they  may  be  put  into  the  perspective  plan 


318  MODEEN  PERSPECTIVE. 

by  the  methods  just  described.  In  this  case  the  points 
and  lines  in  which  the  several  lines  and  planes  cut  the 
plane  of  projection  are  their  initial-points  and  lines,  and 
may  be  set  off  by  scale.  The  points  a  a,  at  which  the 
eaves  of  the  building  in  the  figure,  for  example,  pierce 
the  plane  perspective,  are  equally  far  above  the  Ho- 
rizon, and  on  a  level  with  the  top  of  the  corner  between 
them. 

465.  Fig.  140,  which  is  a  view  of  the  spire  of  the 
p.  ^^Q  church  of  St.  Stephen's,  Walbrook,  illustrates 
Two  lines  of    the  usc  of  scvcral  perspective  plans,  and  also 

vertical  ^         ^  ^ 

measures,  the  advantage  of  taking  a  perspective  plane 
considerably  in  front  of  the  object  instead  of  in  contact 
with  it.  The  extension  of  the  right-hand  and  left-hand 
vertical  planes  of  the  tower  until  they  cut  the  perspective 
plane  gives  five  initial-lines,  or  lines  of  vertical  meas- 
ures, on  each  side,  all  of  which  are  quite  outside  of  the 
picture,  instead  of  one  in  the  middle  of  it,  as  is  the  case 
when  the  front  corner  is  taken  as  the  line  of  measures. 
These  are  lettered  P,  I^',  etc.,  P,  I^',  etc.  respectively. 
This  entirely  frees  the  picture  from  constructive  lines. 

Setting  the  object  some  distance  behind  the  perspec- 
tive plane  of  course  makes  its  perspective  smaller,  but 
this  may  be  met  by  setting  off  the  dimensions  upon  the 
lines  of  measures  at  a  larger  scale,  which,  when  the 
position  of  the  vanishing-points  remains  unchanged,  is 
equivalent  to  moving  the  plane  of  the  picture  nearer  to 
the  object  itself.  In  Fig.  140,  the  scale  employed  at  B  for 
horizontal  distances  upon  the  ground-line,  and  for  vertical 


THE  PRACTICAL  PROBLEM.  319 

dimensions  in  the  lines  P,  1^,  etc.,  is  double  that  of  the 
elevation  at  A,  from  which  the  dimensions  are  taken. 

466.  The  same  result  may  be  produced  by  employing 
scales  of  vertical  measures  beyond  the  object,  gmaii-scaie 
in  accordance  with  the  theory  of  small-scale  ^^^' 
data  discussed  in  a  previous  chapter  (339).  If  these 
are  set  up  half-way  between  the  initial-lines,  or  lines 
of  vertical  measures,  and  their  corresponding  vanishing- 
points,  the  scale  to  be  used  will  be  half  as  large,  as  in 
the  figure  at  v  v,  v'  v',  etc.,  where  the  heights  set  off  are 
the  same  as  in  the  elevation  alongside. 

If  both  scales  of  height  are  used,  as  in  the  figure,  one 

on  the  right  of  the  picture  and  another,  at  half  The  vanish- 
ing-point 
the  scale,  on  the  left,  the  use  of  the  left-hand  unnecessary. 

vanishing-point,  Y^,  may  be  dispensed  with,  the  perspec- 
tives of  the  horizontal  lines  being  put  in  by  drawing  lines 
between  the  corresponding  points  on  the  two  scales. 

467.  In  the  largest  of  the  perspective  plans  em- 
ployed in  the  figure,  below  the  picture,  advantage  is 
taken  of  the  fact  that  the  two  sides  of  the  tower 
are  exactly  alike,  to  dispense  also  with  the  use  of  the 
point  D^.  The  points  ascertained  upon  the  left-hand 
side  by  means  of  the  left-hand  point  of  distance,  D^,  are 
transferred  to  the  right-hand  side  by  means  The  method 
of  the  diagonal  line  directed  towards  V^  the  °^^^^s«°^J«- 
"  vanishing-point  of  45°,"  in  accordance  with  the  prin- 
ciple illustrated  in  Fig.  6,  Plate  II.  (61). 

This  plan  illustrates  also  the  principle  of  Auxiliary 
Horizons  discussed  in   section   365.     But  in-  ^u^iiiary 
stead  of  sinking  the  perspective  plan  in  order  ^°""<^°^' 


320  MODERN   PERSPECTIVE. 

to  prevent  the  angles  of  intersection  from  being  too 
acute,  and  accordingly  putting  the  ground-line  four  or 
five  inches  lower  down,  an  auxiliary  Horizon,  H'  H',  is 
drawn  in  four  or  five  inches  above  the  real  Horizon,  the 
ground-line  being  retained,  and  the  lines  of  the  per- 
spective plan  are  directed  to  the  vanishing-points  and 
points  of  distance  found  upon  this  new  Horizon,  as  in 
Plates  XXII.  and  XXIII. 

468.  Lines  lying  in  an  oblique  plane  can  be  measured 
obM  ue  ^^'  ^^y  i^i^ans  of  a  scale  taken  upon  a  line  of 
planes.  mcasures  which  is  the  initial-line  of  the  plane 
in  which  they  lie,  just  as  well  as  lines  lying  in  hori- 
zontal or  in  vertical  planes.  This  line  will  of  course  be 
parallel  to  the  horizon  of  the  plane  drawn  through  the 
vanishing-points  of  two  of  its  elements  (398).  If  the 
lines  are  parallel  to  the  picture  they  will  be  parallel 
both  to  this  horizon  and  to  tliis  trace  or  line  of  meas- 
ures, and  dimensions  taken  upon  the  line  of  measures 
may  be  transferred  to  the  perspective  line  by  drawing 
parallel  lines  to  any  point  upon  the  horizon  as  a  vanish- 
ing-point. Dimensions  taken  upon  the  line  of  measures 
may  be  transferred  to  lines  lying  in  the  oblique  plane 
and  inclined  to  the  picture,  as  before,  by  means  of  a 
point  of  distance  taken  upon  the  horizon  of  the  oblique 
plane.     See  Fig.  15,  Plate  V.  (111). 

469.  In  the  case  of  lines  that  lie  at  the  intersection 
Choice  of       of  two  planes,  it  is  a  mere  matter  of   con- 

lines  of 

measures.  venienco  whether  they  shall  be  treated  as 
lying  in  one  plane  or  in  the  other.     Either  plane  will 


THE  PRACTICAL  PROBLEM.  321 

do,  its  initial-line  serving  as  a  line  of  measures.  Its 
horizon  will  contain  the  vanishing-point  of  the  line  in 
question,  and  its  points  of  distance  (412).  All  the 
points  of  distance  of  a  line  will  be  at  the  same  distance 
from  its  vanishing-point,  since  the  locus  of  its  points  of 
distance  is  a  circle,  of  which  the  vanishing-point  is  the 
centre  (423),  and  the  radius  the  optical  line. 

470.  These  processes  suffice  not  only  to  give  the  per- 
spective of  every  point  and  line  the  position  ^^^^^^  ^^ 
and  direction  of  which  is  known,  but  to  fur-  °''*^°**'- 
nish  several  methods  by  which  they  can  be  determined. 
Whether  one  or  another  of  the  methods  shall  be  em- 
ployed in  a  given  case  is  a  matter  of  discretion,  in 
which  the  judgment  and  experience  of  the  draughtsman 
must  guide  his  choice.  A  line,  for  instance,  may  be 
determined  in  direction  either  by  fixing  the  position  of 
the  points  in  it,  or  by  finding  its  vanishing-point.  It  is 
sometimes  more  convenient  to  do  one,  sometimes  the 
other.  Whichever  is  employed,  the  other  may  be  used 
to  test  the  accuracy  of  the  result. 

471.  In  completing  a  perspective  drawing,  many 
special  devices  may  be  employed  to  abbreviate  g  ^^.^^ 
labor.  Of  these  the  most  important  are  the  '^^^'''^^• 
different  w^ays  of  dividing  lines  in  a  given  ratio,  the 
different  ways  of  casting  shadows  by  natural  or  by  arti- 
ficial light,  the  use  of  points  of  half-distance  or  quarter- 
distance,  and  the  various  otlier  devices  for  bringing  the 
work  within  small  limits,  Mr.  Adhemar's  scheme  for 
avoiding  the  difficulties  experienced  in  drawing  distant 

21 


322  MODEliN   PEKSPECTIVE. 

objects,  by  the  use  of  inclined  perspective  plans,  with 
the  suggested  modifications,  the  employment  of  lines 
already  existing  as  horizons  of  auxiliary  planes,  and  the 
special  processes  to  be  followed  in  putting  circular  arcs 
into  perspective,  with  the  practical  adjustments  to  be 
made  in  the  results.  It  is  not  necessary  again  to  go 
into  these  details  of  procedure. 

472.  It  is,  however,  worth  while  to  say  that  there  are 
Mechanical  somc  mcchauical  devices  not  mentioned  in  the 
aids.  previous  pages  which  are  of  service  when  van- 
ishing-points are  inconveniently  far  off.  One  of  these 
is  the  employment  of  wooden  or  brass  arcs,  fastened  to 
the  drawing-board,  upon  which  a  T-square  armed  with 
pegs  moves  so  as  to  direct  its  upper  edge  always  to  the 
centre  of  the  arc,  as  a  vanishing-point.  Another  way  to 
effect  the  same  thing  is  to  cut  a  curve  upon  the  handle 
of  the  T-square,  and  move  it  upon  pins  driven  into  the 
drawing-board.     See  Fig.  141. 

473.  A  third  device  is  to  cover  the  paper  with  a  net- 
work of  lines,  drawn  in  pencil,  and  directed  towards  the 
principal  vanishing-points,  as  a  guide  in  sketching.  A 
useful  variation  of  this  is  a  paper  carefully  ruled  in  ink, 
to  be  used  in  sketching  upon  paper  laid  over  the  lines, 
and  thin  enousrh  for  them  to  be  seen  throuoh  it. 

474.  It  may  be  added  that  the  simplest  way,  in  prac- 
tice, to  obtain  the  position  of  the  station-point  in  plan, 
and  the  direction  of  the  principal  right-hand  and  left- 
hand  lines,  E  and  L,  when  the  principal  vanishing-points, 
V^  and  V^,  have  been  assumed  (439),  is  to  drive  pins 
into  the  drawing-board  at  the  vanishing-points,  and  then 


THE  PRACTICAL  PKOBLEM.  323 

to  hold  two  T-squares  at  right  angles  to  one  another,  and 
move  them  upon  these  pins.  The  right  angle  in  which 
they  meet  will  of  course  sweep  the  board  in  a  semi- 
circle, and  if  arrested  at  the  point  where  the  two  edges 
of  the  T-squares  have  the  required  direction,  will  give 
the  station-point. 


PHOEBE   A.    HEARST 
ARCHITECTLTRAL  LIBRARY 


APPENDIX. -NOTES. 


It  may  perhaps  help  to  make  the  main  facts  of  Oblique  or  Two 
Point  Perspective  and  of  Parallel  or  One  Point  Perspective  seem 
as  simple  as  they  really  are,  if  they  are  presented  in  their  simplest 
form.  Plate  XXVIII.  accordingly  shows  several  typical  build- 
ings, with  gables  and  dormers,  hips  and  valleys,  drawn  in  both  of 
these  ways,  with  perspective  plan,  perspective,  and  bird's-eye 
view.  The  lines  and  planes  in  the  plan  and  in  the  bird's-eye 
view,  and  their  vanishing-points  and  horizons,  are  lettered  in 
accordance  with  the  notation  used  in  this  book.     See  Note  VI. 

Note  I.  —  Oblique  Perspective. 

Fig.  142  shows  a  building  in  Two  Point  Perspective,  the  angle 
a  at  the  top  of  the  diagram  showing  the  attitude,  or  the  angle 
which  the  building  makes  with  the  plane  of  the  picture.  The 
position  of  the  Station  Point,  in  front  of  the  picture,  is  shown  in 
its  revolved  position  at  S',  the  right  and  left  hand  vanishing-points 
being  taken  at  V^  and  V''.  The  position  of  S'  is  determined  by 
lines  drawn  parallel  to  those  at  a,  from  V^  and  V^.  This  gives 
D"'  and  D^,  the  points  of  distance,  and  the  position  of  V^,  the 
Centre,  and  of  V^  and  V^  the  vanishing-points  of  diagonals,  in 
the  ])lane  of  the  picture,  upon  the  Horizon.  The  Optical  lines, 
11°  and  L*^,  by  which  these  are  determined,  lie  really  in  a  hori- 
zontal plane  extending  from  the  horizon  H  R  L  to  the  Station- 
Point,  in  the  air  in  front  of  the  picture.  This  plane  is  shown 
revolved  into  the  plane  of  the  picture  about  the  horizon  H  RL. 

The  vertical  horizons  H  R  Z  and  H  L  Z  being  then  drawn,  the 
vanishing-points  V*^,  V**'  and  V^  V^'  are  found  upon  them  by 
layin«'  off  the  angle  ^  at  D^  and  D^.    The  inclined  horizons  H  R  N 


326  MODERN   PERSPECTIVE. 

and  H  R  N',  H  LM  and  H  L  M',  are  then  drawn,  and  the  vanishing- 
points  V^  and  V%  V*^  and  V**',  found  at  their  intersection.  Just 
as  V^  and  the  V^'  come  on  the  horizon  H  X  Z,  above  and  below 
V^,  so  the  vanishing-points  V*^  and  V**'  are  found  in  the  horizon 
H  Y  Z,  above  and  below  V^. 

It  is  to  be  noticed  that  the  position  of  V^,  and  hence  of  H  Y  Z, 
V^  and  V*',  can  be  got  by  drawing  a  construction  line,  otherwise 
meaningless,  through  V"  and  V^  or  through  V"'  and  V%  and  that 
in  like  manner  V^  lies  where  a  line  through  V*^  to  V',  or  V"'  to 
V",  cuts  the  Horizon.  It  is  sometimes  convenient  to  determine 
these  points  in  this  way. 

Note  IL  —  .^°  Perspective. 

Fig.  143  exhibits  the  characteristic  peculiarities  of  45°  Perspec- 
tive :  the  diagram  is  symmetrical  both  about  the  Horizon  H  R  L 
and  about  the  Vertical  Horizon  H  X  Z  ;  V^  V^  and  V*^'  lie  at  an 
infinite  distance  in  the  plane  of  the  picture  ;  H  R  N  is  parallel  to 
H  L  M'  and  to  Q,  and  H  R  N'  to  H  L  M  and  to  Q';  V^  and  V*'  co- 
incide, both  being  at  the  Centre  ;  and  D^  and  D^  are  equidistant 
from  it,  being  about  two-fifths  of  the  way  from  V°  to  V^  or  V^ 

It  is  to  be  noticed  that  D^  and  D''  come  just  where  the  corners 
of  an  octagon  would  come  if  cut  from  a  square  whose  side  was  as 
long  as  the  line  from  V^  to  V^. 

These  conditions  make  it  easier  to  draw  a  building  when  it 
makes  an  angle  of  45°  with  the  picture  than  when  it  is  in  any  other 
position. 

These  conditions — viz.:  that  in  45°  perspective  V^  coincides 
with  V^  and  comes  half  way  between  V^  and  V'',  and  that  while 
one  diagonal  of  a  horizontal  square  is  directed  toward  this  point, 
the  other,  being  directed  toward  the  infinitely  distant  V^,  is  paral- 
lel to  the  Horizon,  that  is  to  say,  horizontal  —  make  it  very  easy 
to  put  any  number  of  such  squares  into  perspective.  As  the  plans 
of  most  buildings  are  laid  out  on  a  modulus  of  equal  parts,  that  is 
to  sny,  are  composed  of  squares,  it  is  accordingly  very  easy  to  draw 
a  perspective  plan  of  such  buildings  and  hence  to  draw  the  per- 


APPENDIX.  327 

spectives  of  the  buildings  themselves.  The  scale  of  the  vertical 
lines  is  the  same  at  any  point  as  that  of  the  horizontal  diagonal 
of  a  square  at  that  point,  the  nearer  squares  coming  out,  of  course, 
larger  than  the  more  distant  ones. 

Let  us  suppose  that,  as  in  Fig.  144,  we  have  a  building  100  feet 
long,  50  feet  wide,  with  a  tower  50  feet  square  and  a  porch  25  feet 
by  50,  25  feet  high  up  to  the  eaves,  50  feet  to  the  ridge,  and  100 
feet  to  the  top  of  the  tower,  and  that  the  porch  walls  are  half  as 
high  as  those  of  the  church. 

V^  and  V^  being  taken  as  far  apart  as  is  convenient  upon  a  line 
taken  for  the  Horizon,  and  V^  half  way  between  them,  the  three 
whole  squares  and  one  half  scpiare  of  the  perspective  plan  can 
be  drawn  anywhere  below  it  at  any  convenient  scale,  the  sides  of 
the  squares  being  directed  toward  V^  and  V%  one  of  the  diagonals 
toward  V^,  and  the  other  drawn  horizontally.  These  may  be 
taken  to  represent  the  perspective  of  squares  that  measure  50  feet 
on  a  side.  The  perspective  of  the  building  can  then  be  drawn 
directly  above  the  plan,  the  lower  end  of  the  nearest  front  corner 
being  taken  just  over  the  corner  of  the  nearest  square,  and  the 
other  vertical  lines  drawn  above  the  other  corners.  The  vertical 
line  that  passes  through  the  gable  is  drawn  through  the  middle  of 
the  left-hand  side  at  a  point  determined  by  drawing  a  line  through 
the  middle  of  the  building  in  the  perspective  plan  below.  The 
height  of  50  feet  is  then  to  be  laid  off  on  the  front  corner  of  the 
building.  The  length  of  50  feet  at  that  point  is  determined  as 
follows  :  From  the  point  below  it,  a,  in  the  perspective  plan,  is 
drawn  the  diagonal  of  a  third  square  from  a  to  h.  This  diagonal 
is  of  course  drawn  to  the  scale  at  which  the  front  corner  of  the 
building  is  to  he  drawn,  heing  equally  distant  from  the  plane  of 
the  picture.  It  is  the  diagonal  of  a  square  whose  side  is  50  feet. 
The  length  of  that  side,  that  is  to  say,  the  length  of  a  line  50  feet 
long  at  that  point,  is  found  by  erecting  upon  the  line  a  &,  as  a 
hypothenuse,  a  half-square  ah  c.  The  line  &  c  is  obviously  a  line 
50  feet  long,  at  the  scale  in  question. 

If  now  in  the  perspective  above  we  lay  off  this  distance  upon  the 
front  corner  from  a' to  a'\  half  this  height  will  give  the  required 
25  feet  for  the  height  of  the  eaves,  and  the  other  half  the  addi- 


328  MODERN   PERSPECTIVE. 

tional  25  feet  for  tlie  height  of  the  ridge.  Doubling  this  line 
gives  the  height  of  the  tower,  and  halving  it  the  height  of  the 
porch  walls.  These  are  all  the  data  required  for  completing  the 
drawing. 

Here,  as  in.  most  cases,  the  scale  of  the  perspective  is  on  the 
whole  smaller  than  the  scale  at  the  nearest  corner,  and  as  it  is 
generally  convenient  to  make  the  scale  at  this  corner  the  same  as 
that  of  the  orthographic  elevation,  the  result  is  that  the  details 
have  to  be  drawn  considerably  smaller  in  the  perspective  than  in 
the  geometrical  elevations.  This  can  be  obviated  by  taking  the 
Perspective  Plane  somewhere  in  the  middle  of  the  building,  or, 
as  is  sometimes  done,  using  one  of  the  further  corners  instead  of 
the  front  corner  as  a  line  of  vertical  measures.  In  this  case  the 
half  of  the  building  which  is  seen  comes  in  front  of  the  perspective 
plane  instead  of  behind  it,  and  the  details  are  drawn  on  a  scale 
somewhat  larger  than  that  of  the  elevations  instead  of  smaller. 

The  relations  just  described  make  it  easy  to  effect  this  in  the 
case  of  a  square  building,  such  as  a  monument  or  tower,  set  at 
an  angle  of  45°. 

This  is  illustrated  in  Fig.  145,  which  shows  such  a  structure  in 
elevation  at  A.  At  B  it  is  shown  in  diagonal  elevation,  a  view 
which  is  often  worth  the  trouble  to  make  in  order  to  see  how  a 
design  will  appear  in  its  most  unfavorable  aspect.  It  is  of  course 
easily  made  by  laying  off  upon  horizontal  lines  the  "diagonals" 
of  the  horizontal  distances  from  the  centre  instead  of  the  actual 
distances.  If  now  at  the  points  thus  ascertained  only  the  vertical 
elements  of  the  outline  are  drawn,  as  shown  at  C,  it  is  easy  to 
construct  the  required  perspective  drawing  within  the  limits  thus 
determined.  All  that  is  necessary  is  to  draw  a  Horizon  at  any 
convenient  height  and  to  take  upon  it  three  equidistant  points 
V^,  Y^,  and  V^.  Pains  must  be  taken  not  to  have  V^  come  exactly 
on  the  axis  of  the  building.  If  now  from  the  extremity  of  the 
vertical  outlines  thus  determined  lines  are  drawn  from  V^  and 
V^,  the  points  where  they  intersect  will  determine  the  perspec- 
tive of  the  front  corner.  The  result  will  be  a  true  perspective  of 
the  building,  and  the  outline  will  be  drawn  at  the  same  scale  as 
the  elevation,  as  appears  in  the  figure. 


APPENDIX.  329 

Note  III.  —  Parallel  Perspective. 

Fig.  146  exhibits  in  a  somewhat  simpler  manner  than  Plate  VI. 
the  distinctive  features  of  One  Point,  or  Parallel,  Perspective. 
Here  again  the  diagram  is  symmetrical  about  both  the  vertical 
and  the  horizontal  horizons  that  pass  through  the  centre,  V^.  The 
vertical  planes  are  either  parallel  or  normal  to  the  plane  of  the 
picture,  and  the  inclined  planes  of  the  roof  are  either  normal,  as 
in  the  case  of  the  planes  C  D  and  C  S,  or,  as  in  the  case  of  the 
planes  K  A  and  K  A',  they  have  their  horizontal  element  parallel 
to  the  picture,  so  that  their  horizons,  H  K  A  and  H  K  A',  are  hori- 
zontal and  parallel  to  the  Horizon  H  C  K  (or  H  R  L) ;  the  diago- 
nals X  and  Y  coincide  in  direction  with  R  and  L,  and  the  lines  of 
the  hips  and  valleys  P  and  P',  Q  and  Q',  with  M  and  M',  N  and  N'. 

Note  IV.  —  The  Inverse  Process. 

Besides  the  three  devices  mentioned  in  the  text  there  are  three 
other  ways  of  obtaining  an  elevation  from  a  perspective  drawing 
or  photograph.  The  problem  in  every  case  is  to  find  D^  and 
D^  when  Y^  and  Y^  are  given,  some  third  point  being  also  sup- 
plied. It  is  plain  from  the  relations  shown  in  Fig.  139  A,  Plate 
XXYII.,  and  again  in  Fig.  148,  that  if  Y^  and  Y^  are  given,  and 
also  either  S',  Y°,  D^,  or  D^,  the  other  three  points  can  easily  be 
obtained. 

I.  The  first  case  is  that  in  which  the  Centre  Y*^  is  indicated  by 
some  object  given  in  Parallel  Perspective.  This  case  has  already 
been  illustrated  in  Fig.  Ill  A,  Plate  XXIY.,  and  discussed  in 
Section  384,  page  250. 

The  whole  procedure  is  shown  in  Fig.  148,  Plate  XXIX.  S'  is 
obtained  from  Y°,  and  D^  and  D''  from  S'.  Lines  drawn  from 
D^  determine,  upon  the  Ground  Line,  T  R  L,  the  points  a,  b,  c,  d, 
€,/,  g,  and  h,  which  give  the  real  width  of  the  piers  and  arches  on 
the  right-hand  side  of  the  building,  on  the  scale  of  the  nearest 
corner.  In  like  manner  lines  drawn  from  I)^  determine  the  points 
i,j,  and  k,  which  give  the  horizontal  dimensions  of  the  left-hand 
side,  on  the  same  scale.  The  points  /,  m,  n,  o,  p,  q,  r,  s,  t,  u,  and  y, 
taken  from  points  on  the  front  corner  itself,  give  the  vertical 


330  MOHERN   PEKSPECTIVE. 

dimensions.     From  these  the  two  elevations  can  be  constructed, 
as  is  done  in  Fig.  154. 

II.  The  second  case  is  that  in  which  V^  is  indicated  by  a  hori- 
zontal square,  the  diagonal  of  which  is  directed  to  V^  as  its 
vanishing-point.  In  Fig.  149  such  a  square  is  found  by  taking 
equal  distances  upon  the  riglit  and  left  hand  cornices,  each  in- 
cluding six  modillions.  By  drawing  a  circle  of  which  the  line 
yL  yR  ig  the  diameter,  and  drawing  a  line  from  the  top  of  the 
circle  through  V^,  the  position  of  S'  is  determined,  as  has  been 
already  explained  in  Section  385.  D^  and  D^  are  then  easily  found, 
as  in  the  preceding  example. 

III.  The  third  case  is  that  in  which  V^  is  given  by  the  hips 
and  valleys  of  which  it  is  the  vanishing-point,  Fig.  150.  V^  is 
found  upon  the  Horizon  directly  below  it,  and  the  problem  becomes 
the  same  as  in  the  previous  case. 

IV.  The  fourth  case  is  that  in  which  a  square  or  half-square 
occurs  in  a  vertical  plane,  or  is  given  by  a  circle  or  half-circle,  as, 
for  example,  by  a  circular  window,  or  by  a  semicircular  arch, 
as  in  Fig.  151.  The  diagonals  of  these  parallelograms  have  their 
vanishing-points  at  V"i  or  V"i,  on  H  R  Z,  the  vertical  horizon  of 
the  plane  in  which  they  lie.  The  true  inclination  of  these  lines 
is  known,  being  the  angle  which  they  make  with  the  horizontal 
plane,  the  angle  which  we  have  called  ^.  D^  is  easily  found  by 
laying  off  the  complement  of  this  angle  at  V**i  or  V*'^,  and  draw- 
ing a  line  to  the  Horizon. 

The  simplest  way  to  get  D^  is  to  lay  off  on  the  Horizon,  from 
V^,  the  distance  V^  Vi,  or  twice  the  distance  V^  D"j,  that  is  to 
say,  to  revolve  into  the  plane  of  the  picture  the  base  of  the  optical 
triangle,  S  V»  V^,  or  S  V^  V\ 

If  the  vertical  square  or  half-square  were  on  the  left-hand  side 
of  the  building,  V^i  or  V^j  would  be  employed  to  obtain  D^ 

The  point,  V**,  on  H  E,  Z,  indicates,  as  usual,  the  vanishing-point 
of  the  lines  of  the  gable  and  of  the  steepest  line  of  the  roof. 

Of  course  any  other  feature  whose  shape  was  known  would  give 
the  angle  ^  as  w^ell  as  a  square  or  semicircle. 

V.  The  fifth  case  gives  a  point  of  distance,  D^  or  D^  directly. 
Fig.  152.  Here  also  a  vertical  square  occurs  in  a  vertical  perspec- 
tive plane.      If  a  horizontal  line  is  drawn  from  one  end  of  its 


APPENDIX.  331 

vertical  side  as  long  as  that  side,  this  line  and  the  perspective  of 
the  horizontal  side  will  be  the  perspective  of  the  sides  of  an 
isosceles  triangle  lying  in  a  horizontal  plane.  The  vanishing- 
point  of  the  base  of  this  triangle  will  be  the  required  point  of 
distance.  This  isosceles  triangle  is  half  of  a  horizontal  square, 
and  its  base  is  the  diagonal  of  the  square.  S',  the  revolved  posi- 
tion of  the  Station  point,  and  the  other  point  of  distance  are  then 
easily  found. 

If  the  isosceles  triangle,  as  in  the  figure,  lies  so  near  the  Hori- 
zon as  to  be  very  much  foreshortened,  the  angles  coming  out  too 
acute  for  accurate  draughtsmanship,  it  can  be  re-drawn  at  a  lower 
level,  as  shown. 

This  is  the  same  principle  that  is  illustrated  in  Plate  XXIV., 
Fig.  113,  and  discussed  in  Section  386. 

VI.  The  sixth  case,  illustrated  in  Fig.  153,  also  gives  a  point 
of  distance  directly,  and  again  employs  a  vertical  square,  using  it 
just  as  the  horizontal  square  was  used  in  Fig.  152.  The  point  of 
distance  of  R,  the  horizontal  side  of  the  square,  is  again  the  first 
point  determined.  But  as  the  square  of  which  this  line  is  a  side 
is  now  a  vertical  square,  this  point  of  distance,  instead  of  being  on 
the  Horizon,  H  R  L,  lies  in  the  vertical  horizon,  H  R  Z.  Either 
half  of  the  square  is  an  isosceles  triangle,  the  base  of  which  is  the 
diagonal  of  the  square,  and  the  vanishing-point  of  this  diagonal, 
in  the  vertical  horizon  H  R  Z,  is  accordingly  the  vanishing-point 
of  the  base,  or  point  of  distance,  D^.  Lines  drawn  from  this 
point  through  the  lower  ends  of  the  piers  determine  upon  the 
vertical  trace  TRZ  the  points  a,  h,  c,  d,  e,/,  g,  and  h,  which  give 
the  real  width  of  the  piers  and  arches  on  the  right-hand  side  of 
the  building,  just  as  these  were  determined  on  the  Ground  Line, 
TRL,  inFig.  148. 

D^  on  the  Horizon  H  R  L  is  of  course  just  as  far  from  V^  as  is 
D^  on  H  R  Z,  both  distances  being  equal  to  the  length  of  the 
Optical  line  R°.  Both  points  are  situated  on  a  circle  of  which 
V^  is  the  centre  and  R*'  the  radius,  and  which  is  the  locus  of  D^. 
S',  D^,  V^,  and  V^  can  then  easily  be  found,  if  wanted. 


332  MODEKN    PEHSPECTIVE. 

Note  V.  —  Shadoivs  hy  Artificial  Liglii. 

Fig.  155  illustrates  the  proposition  that  if  an  auxiliary  line,  or 
ray,  be  drawn  through  a  source  of  divergent  rays  as  an  Apex, 
parallel  to  a  given  system  of  lines,  the  shadows  of  tliose  lines  upon 
any  plane  will  diverge  from  the  point  where  the  auxiliary  line 
pierces  the  plane.  This  has  already  been  illustrated  in  Plates 
XVII.,  XVIIL,  and  XIX.  In  the  figure  the  rays  of  light  diverge 
from  A  as  their  Apex,  the  shadows  of  the  vertical  lines  Z  from 
Az,  and  the  shadows  of  the  inclined  lines  M  from  Ajj. 

In  Fig.  155  the  source  of  light,  A,  is  so  low  down  and  Y",  the 
vanishing-point  of  the  inclined  lines,  is  so  high  up,  that  the  point 
Am,  vvhere  the  auxiliary  line  strikes  the  horizontal  plane,  lies 
between  the  ground  line  and  the  Horizon.  But  it  may  happen 
that  the  source  of  light  is  so  high  above  the  ground  plane,  or  the 
inclination  of  the  given  lines  so  slight,  that  the  auxiliary  line 
pierces  the  plane  of  the  picture  before  striking  the  ground  plane. 
In  that  case  the  Apex  Am  will  be  in  front  of  the  plane  of  the 
picture  and  may  even  be  so  far  in  front  of  it  as  to  be  behind  the 
spectator. 

This  is  illustrated  in  Fig.  156.  In  the  orthographic  plan  below 
are  shown  the  plane  of  the  picture,  pp,  and  the  station-point  in 
front  of  it,  S.  The  optical  line  C°,  normal  to  the  picture,  gives 
the  position  of  the  Centre  Y^,  and  the  optical  line  E°,  which  is 
the  horizontal  projection  of  M",  the  optical  line  of  the  given 
system,  gives  the  position  of  Y^  and  the  position  of  D^.  In  the 
perspective  above,  the  angle  laid  off  at  D^  on  the  Horizon  gives 
the  position  of  the  vanishing-point  Y*^.  If  now  the  position  of 
the  source  of  light  is  assumed  to  be  at  A,  and  its  projection  on  the 
ground  plan  to  be  at  Az,  the  auxiliary  line  M  can  be  drawn 
through  Y'"  and  A,  and  its  projection  R  on  the  ground  plane 
through  Y^  and  Az.  H  and  M  will  both  lie  in  a  vertical  plane 
R  Z  and  their  vanishing-points  \\\\\  be  in  its  horizon,  H  RZ,  and 
their  initial -points  in  its  trace,  T  R  Z.  R  will  pierce  the  plane  of 
the  picture  at  its  initial-point  P,  where  it  cuts  the  ground  line, 
and  if  produced  in  front  of  the  picture,  as  shown  in  the  ortho- 
graphic plan  below,  it  will  lie  in  the  ground  plane,  in  a  direction 


APPENDIX.  333 

parallel  to  the  optical  line  R**.  The  auxiliary  line  M  will,  if  it 
also  is  continued  in  front  of  the  picture,  descend  from  its  initial- 
point  1"  until  it  meets  the  line  R  at  Am-  This,  as  before,  is  the 
apex  from  which  diverge  the  shadows  of  the  lines  M  cast  upon  the 
ground  plane. 

The  point  Am  is  easily  found,  since  the  trace  T  E,  Z  is  one  side 
of  a  vertical  right  triangle,  extending  in  front  of  the  picture,  of 
which  the  line  R,  produced,  is  the  base  and  the  line  M,  produced, 
is  the  hypothenuse.  The  length  of  the  line  R  is  found  by  revolv- 
ing this  triangle  into  the  plane  of  the  picture  around  the  trace 
T  R  Z,  as  shown,  making  the  angle  at  the  base  equal  to  /3.  That 
is  to  say,  the  hypothenuse  M  in  its  revolved  position  is  drawn 
parallel  to  the  line  D^  V",  which  is  M^  in  its  revolved  position. 

It  is  to  be  observed  that  the  auxiliary  line  M  taken  through  A 
really  ascends  from  1"  to  V",  since  it  makes  the  angle  /3  with  the 
ground  plane.  But  it  seems  to  descend,  the  perspective  of  its  upper 
end  V**  coming  lower  down  on  the  paper  than  that  of  its  lower 
end  1"^.  This  often  happens  when  one  looks  up  at  an  ascending 
line  at  an  angle  steeper  than  that  of  the  line  itself. 

The  apex  Am  being  behind  the  spectator,  the  shadows  of  the  lines 
M,  cast  upon  the  ground  plane,  which  really  diverge  from  this  apex, 
seem  to  converge  upon  a  false  apex  Am',  in  front  of  the  spectator. 

This  false  apex  may  be  found,  as  in  Fig.  80,  Plate  XVII.,  by 
passing  through  the  apex  Am  two  horizontal  lines  diverging  from 
it,  one  normal  to  the  plane  of  the  picture  and  the  other  passing 
directly  under  the  station-point.  These  lines  lie  in  the  ground 
plane.  Calling  them  respectively  C  and  R',  the  portions  in  front 
of  the  plane  of  the  picture  as  shown  in  the  orthographic  plan  will 
pierce  the  plane  of  the  picture  at  I^  and  1%  on  the  ground  line, 
and  the  perspectives  of  the  portions  behind  the  picture  will  extend 
from  1°  to  V*^  and  from  P'  to  V%  respectively.  These  lines  point 
to  the  false  apex  Am'- 

But  as  M  and  R  also  diverge  from  the  apex  Am  as  their  real 
apex,  the  perspectives  of  these  lines  also  are  directed  toward  the 
false  apex,  and  Am'  cnn  be  found  at  once,  without  making  any 
orthographic  plan  at  all.  It  is  at  the  intersection  of  M  and  R, 
extended,  just  as  the  real  apex  is  found  at  the  intersection  of  M 

155. 


334 


MODERN   PERSPECTIVE. 


A,  A' 


If  A,  the  apex  of  diverging  rays,  is  so  situated  that  the  apex  of 
shadows,  Am,  though  in  front  of  the  picture,  is  neither  in  front  of 
the  spectator  nor  behind  him,  the  line  C  being  just  as  long  as  the 
Axis  C^,  then  the  perspectives  of  the  diverging  shadows  will  be 
parallel,  just  as  the  lines  2,  7,  and  4  and  lines  5,  8,  and  10  are  in 
Fig.  77,  and  their  false  apex  will  be  at  an  infinite  distance.  This 
is  illustrated  in  Fig.  157.  It  will  be  observed  that  under  these 
conditions  the  apex  A  conies  just  as  far  above  the  ground  line  as 
the  vanishing-point  V"  is  above  the  Horizon,  and  that  M,  R,  and 
the  shadows  of  M  on  the  ground  plane  are  all  parallel  also. 

Under  these  circumstances  W  becomes  parallel  to  the  picture 
and  is  replaced  by  K. 

Note  VI.  —  The  Perspective  Alphahet. 

The  Notation  employed  in  this  book  is  exhibited  in  the  follow- 
ing Table.  The  words  in  parenthesis  explain  the  significance  of 
the  letters,  and  make  it  easy  to  remember.  Most  of  these  letters 
stand  for  lines.    Those  which  indicate  points  are  marked  thus  X. 

{Altitude)  Inclined  lines,  sloping  up  or  down,  in  vertical 

normal  planes. 
An  Apex.    The  point  at  which  convergent 

lines  really  meet. 
A  False  Apex,  or  point  toward  which  diver- 
gent lines  seem  to  converge. 
Other  apexes  and  false  apexes. 

{Base)  Lines  parallel  to  the  base  of  the  isosceles 

triangles,  having  the  point  of  distance  for 
their  vanishing-points.     V^  =  D. 

{Centre)  Normal  Lines,  having  their  vanishing-points 

at  the  Centre  VC. 

(Dexter)  Lines  parallel  to  the  picture  sloping  down  to 

the  right. 

(Distance)  A  point  of  distance,  which  is  a  point  at  the 

same  distance  from  the  vanishing-poiut 
that  the  vanishing-point  is  from  the  station- 
point.  S  V  =  V  D. 
Points  of  distance  of  the  systems  R,  L,  M,  etc. 
It  is  sometimes  convenient  to  use  these  let- 
ters to  designate  the  main  lines  in  Three 
Point  Perspective. 

(Horizon)  A  Horizon,  the  infinitely  distant  line  where 

the  parallel  planes  of  any  system  seem  to 
meet,  or  its  perspective  in  the  plane  of 
the  picture. 


X    A 


X    A' 


B,  B' 
B 


X    D 


DR,  Di-,  DM,  etc. 
E,  F,  G 


IR,  l^,  etc. 

K 

(Instead  of  H) 

L,L' 

{Left  hand) 

M,  M' 

N,N' 

APPENDIX.  335 


H',  H",  etc.  Horizons  of  auxiliary  planes. 

H  R  L,  H  R  Z,  etc.  The  Horizons  of  the  planes  R  L,  R  Z,  etc ; 

or  their  perspectives.     Horizons  are  uadi- 

cated  thus  : . 

H  H  =  H  R  L  The    Horizon ;    the    horizon    of    horizontal 

planes. 
X    I  {Initial)  An  Initial  point ;  the  point  where  an  inclined 

line  behind  the  plane  of  the  picture  pierces 
it ;  the  first  or  nearest  pcint  of  such  a  line. 
The  Initial  points  of  the  lines  R  L,  etc. 
Horizontal  lines  parallel  to  the  plane  of  the 

picture. 
Horizontal  lines  inclined  to  the  plane  of  the 

picture  and  going  back  to  the  Left. 
Oblique  lines  going  up,  or  down,  and  back  to 

the  Right. 
Oblique  lines  going  up,  or  down,  and  back  to 
the  Left. 
O  {Optical)  This  letter  designates  an  optical  line,  that  is, 

the  element  of  a  system  which  passes 
through  the  eye,  or  station-point.  It  is  in 
front  of  the  picture,  which  it  pierces  at  the 
perspective  of  the  vanishing-point  of  the 
system.  Its  length  is  the  distance  of 
the  vanishing-point  from  the  station-point. 
RO,  LP,  etc.  The  optical  lines  of  the  systems  R,  L,  etc. 

CO  The  Axis;   the  optical  hue  of  the  system  C, 

normal  to  the  plane  of  the  picture.     Its 
length  is  the  distance  of  the  eye  from  the 
nearest  point  of  the  picture,  that  is,  from 
the  Station-point  to  the  Centre. 
P  P  The  Perspective  plane,  or  Plane  of  Measures, 

situated  near  the  object. 
pp  The  Plane  of  the  Picture,  situated  near  the 

spectator,    parallel    to    the    Perspective 
Plane. 
(When  the  object  is  small,  or  is  a  model  of  the  real  object,  P  P  coincides 
vnthpp.) 
p,  P'  Oblique  lines,  such  as  hips  and  valleys,  lying 

in  or  parallel  to  the  intersections  of  oblique 
planes,  and  nearly  normal  to  the  picture. 
Q,  Q'  The  same,  when  nearly,  or  quite,  parallel  to 

the  picture. 
R,  R'  (Right  hand)      Horizontal  lines  inclined  to  the  plane  of  the 

picture  and  going  back  to  the  Right. 
S  (Sinister)  Lines  parallel  to  the  picture,  sLping  down  to 

the  Left. 
S  (Spectator)  The  Station-point,  or  position  of  the  eye,  in 

the  air  in  frcnt  of  the  picture. 


336  MODERN    PERSPECTIVE. 

X     S',  S",  etc.  The  position  of  the   Station-point  when  re- 

volved into  the  plane  of  the  picture. 
T  ( Trace)  A  Trace,  or  initial-line  ;  the  line  in  which  an 

inclined  plane  behind  the  plane  of  the 
picture  intersects  it.    Traces  are  indicated 

thus : . 

T',  T",  etc.  Auxiliary  traces. 

T  R  L,  T  R  Z,  etc.  The  traces  of  the  planes  R  L,  R  Z,  etc. 

T  {T-square)  Lines  normal  to  a  given  system  of  planes ; 

their  axes. 
TRLj  TLM,  etc.  Lines  normal  to  the  planes  R  L,  L  M,  etc. 

X     V  A  vanishing-point ;  the  infinitely  distant  point 

where   the  lines  of  any  system  seem  to 
meet  ;   or  its  perspective  in  the  plane  of 
the  picture. 
\^-,  V'',  yM,  etc.  The  vanishing-points  of  the  systems  R,  L,  M 

etc.,  or  their  perspectives. 
X     V^  "The  Vanishing-point  of  diagonals,"  or  "of 

Forty-Five  Degrees." 
X     V,  V"  _  Vanishing-points  of  auxiliary  lines. 

X  (    [X     )  "  Diagonal  "  lines,  or  "  45^  "  lines  ;  horizon- 

tal lines  nearly  or  quite  normal  to  the 
picture,  and  in  the  direction  of  one  diago- 
nal of  a  square,  the  sides  of  which,  R  and 
L,  are  at  right  angles  with  one  another. 
Y  Similar  lines  in  the  direction  of  the  other 

diagonal,  and  nearly  or  quite  parallel  to 
the  picture. 
Z  {Zenith)  Vertical  lines,  parallel  to  the  plane  of  the 

picture,  and  having  their  vanishing-points 
in  the  Zenith  and  Nadir. 
Z  Z  =  C  Z  The  principal  vertical  horizon  ;  the  horizon 

of  vertical  planes  normal  to  the  picture. 
The  Letters  J,  U,  and  W  are  not  employed. 

G  L,  or  i?  Z  These    letters    are    used  to  designate    the 

Ground  Line,  or  trace  of  the  ground  plane. 

Planes  are  designated  by  the  letters  which  indicate  their  two  principal  elements, 

viz.  :  The  horizontal  element  and  the  line 

of  steepest  slope  ;  as  R  N,  C  S,  L  Z,  etc. 

Finite  lines  are  designated  by  small  capitals  :  r,  l,  c,  y,  z,  etc. 

Points  are  marked  thus  X,  and  are  designated  by  small  letters,  thus  :  cr,  6,  etc., 

TO,  n,  etc. 
Letters  denoting  lines  revolved  into  the  plane  of  the  picture  are  enclosed  in 
parentheses,  thus :  —  (R*'). 


BUILDING  SUPERINTENDENCE. 

A  MANUAL  FOR  YOUNG   ARCHITECTS,    STUDENTS,   AND   OTHERS  INTER- 

ESTED  IN   BUILDING   OPERATIONS  AS  CARRIED  ON 

AT  THE   PRESENT  TIME. 

By  T.   M.  CLARK, 

Fellow  of  the  American  Institute  of  Architects. 

In  one  volume,  square  8vo.     336  pp.     Illustrated  with  194  Plans,  Diagrams, 
etc.    Price  $3.00. 


CONTENTS. 


Introduction. 

The  Construction  of  a  Stone  Church. 

Wooden  Dwelling-houses. 

A  Model  Specification. 


Contracts. 

The  Construction  of  a  Town  Hall. 

Index. 


"This  is  not  a  treatise  on  the  architectural  art,  or  the  science  of 
construction,  but  a  simple  exposition  of  the  ordinary  practice  of  building 
in  this  country,  with  suggestions  for  supervising  such  work  efficiently. 
Architects  of  experience  probably  know  already  nearly  everything  that 
the  book  contains,  but  their  younger  brethren,  as  well  as  those  persons 
not  of  the  profession,  who  are  occasionally  called  upon  to  direct  building 
operations,  will  perhaps  be  glad  of  its  help." 

There  is  hardly  any  practical  problem  in  construction,  from  the  build- 
ing of  a  stone  town-hall  or  church  to  that  of  a  wooden  cottage,  that  is  not 
carefully  considered  and  discussed  here;  and  a  very  full  index  helps  to 
make  this  treasury  of  facts  accessible.  Every  person  interested  in  build- 
ing should  possess  this  work,  which  is  approved  as  authoritative  By  the 
best  American  architects. 

This  volume  has  been  used  for  years  as  a  text-book  in  the  chief  Archi- 
tectural Schools  in  the  United  States. 


BY  THE  SAME  AUTHOR. 


ARCHITECT,    OWNER,    AND    BUILDER 
BEFORE  THE  LAW. 

By  T.  M.   CLARK. 
Square  8vo.  Cloth.    $3.00. 


THE    MACMILLAN    COMPANY, 

66   FIFTH   AVENUE,  NEW  YORK. 


PRACTICAL   BOOKS   FOR   THE   ARCHITECT. 


SAFE   BUILDING. 

By  LOUIS  DeCOPPET  BERG. 
In  two  volumes,  square  8vo.    Illustrated.     Price,  $5.00  each  volume. 


^♦,„  An  edition  of  Vol.  I.  may  also  be  had  in  pocket  form,  in  flexible 
roan,  with  flap.     Price  ^3.00. 


The  author  proposes  to  furnish  to  any  earnest  student  the  opportunity 
to  acquire,  so  far  as  books  will  teach,  the  knowledge  necessary  to  erect 
safely  any  building.  First  comes  an  introductory  chapter  on  the  Strength 
of  Materials.  This  chapter  gives  the  value  of,  and  explains  briefly,  the 
different  terms  used,  such  as  stress,  strain,  factor  of  safety,  centre  of 
gravity,  neutral  axis,  moment  of  inertia,  etc.  Then  follows  a  series  of 
chapters,  each  dealing  with  some  part  of  a  building,  giving  practical  advice 
and  numerous  calculations  of  strength;  for  instance,  chapters  on  founda- 
tions, walls  and  piers,  columns,  beams,  roof  and  other  trusses,  spires, 
masonry,  girders,  inverted  and  floor  arches,  sidewalks,  stairs,  chimneys,  etc. 

These  papers  are  the  work  of  a  practising  architect,  and  not  of  a  mere 
book-maker  or  theorist.  Mr.  Berg,  aiming  to  make  his  work  of  the  great- 
est value  to  the  largest  number,  has  confined  himself  in  his  mathematical 
demonstrations  to  the  use  of  arithmetic,  algebra,  and  plane  geometry. 
In  short,  these  papers  are  in  the  highest  sense  practical  and  valuable. 


CONTENTS. 


Volume  I. 

Chapter 

I.    Strength  of  Materials. 
II.    Foundations. 

III.  Cellar  and  Retaining  Walls. 

IV.  Walls  and  Piers. 
V.    Arches. 

VI.    Floor-beams  and  Girders. 
VII.    Graphical  Analysis  of  Trans- 
verse Strains. 


Volume  II. 

Chapter 

VIII.    The  Nature  and  Uses  of  Iron 
and  Steel. 
IX.    Rivets,  Riveting,  and  Pins. 
X.    Plate  and  Box  Girders. 
XI.    Graphical  Analysis  of  Strains 

in  Trusses. 
XII.    Wooden  and  Iron  Trusses. 
XIII.    Columns. 

Tables.  Index. 


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